from Lemma 2.5.1, and considering the fact that the constant $c_\sigma > 1$ is unknown, the assumption $A_N = o(\sqrt{\log(N)})$ is sufficient for the validation of Condition (2.15) which leads, together with Assumption 2.2.1, to the result of Theorem 2.5.2. The study of the validation of Condition (2.15) for $A_N > \sqrt{\log(N)}$, which may require more sophisticated techniques, is beyond the scope of this paper. Finally, under the assumptions of Theorem 2.5.2 and considering the risk of $\hat{\sigma}_{A_N,m}^2$ based on the empirical norm $\|\cdot\|_n$, we also obtain

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{A_N,m}^2 - \sigma_{A_N}^2 \right\|_n^2 \right] = O \left( \log^\beta(N) (Nn)^{-2\beta/(2\beta+1)} \right). $$

In fact, under Condition (2.15), the estimator $\hat{\sigma}_{A_N,m}^2$ satisfies the results of Theorem 4.1 with $I = [-A_N, A_N]$ and $A_N = o(\sqrt{\log(N)})$, which implies rates of the same order for the two empirical norms.

## 2.6 Adaptive estimation of the diffusion coefficient from repeated observations

In this section, we propose an adaptive ridge estimator of $\sigma^2$ by selecting an optimal dimension from the sample $D_{N,n}$. In fact, consider the estimator $\hat{\sigma}_K^2$ where $\hat{K}$ satisfies:

$$ \hat{K} := \arg \min_{K \in \mathcal{K}} \{ \gamma_{n,N} (\hat{\sigma}_K^2) + \text{pen}(K) \} \quad (2.17) $$

and the penalty function $\text{pen}: K \mapsto \text{pen}(K)$ is established using the chaining technique of [7]. We derive below the risk of the adaptive estimator of $\sigma^2|_I$ when the interval $I \subset \mathbb{R}$ is compact, and the number $N$ of sample paths tends to infinity.

**Theorem 2.6.1.** Suppose that $N \propto n$, $L = \log(N)$, $N \to \infty$ and consider the collection $[\mathbf{B}]$ with

$$ K \in \mathcal{K} = \{2^q, q = 0, 1, \dots, q_{\max}\} \subset \mathcal{M} = \{1, \dots, \lfloor \sqrt{N}/\log(N) \rfloor\}. $$

Under Assumption 2.2.1, there exists a constant $C > 0$ such that,

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2}^2 \right\|_{n,N}^2 \right] \le 34 \inf_{K \in \mathcal{K}} \left\{ \inf_{h \in S_{K+M,L}} \|h - \sigma_{|I|^2}^2\|_n^2 + \text{pen}_1(K) \right\} + \frac{C}{Nn} $$

where the penalty function $\text{pen}_1$ is given by

$$ \text{pen}_1(K) := \kappa \frac{(K+M)\log(N)}{Nn} \quad (2.18) $$

with $\kappa > 0$ a numerical constant.

We deduce from Corollary 2.4.2 and its assumptions that the adaptive estimator $\hat{\sigma}_K^2$ satisfies:

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2}^2 \right\|_n^2 \right] = O \left( (Nn)^{-2\beta/(2\beta+1)} \right). $$

This result is justified since the penalty term is of the same order (up to a log-factor) as the estimation error established in Theorem 2.4.1. If we consider the estimation interval $[-A_N, A_N]$ where $A_N \propto \sqrt{\log(N)}$, then we obtain the following result:

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2}^2 \right\|_{n,N}^2 \right] \le 34 \inf_{K \in \mathcal{K}} \left\{ \inf_{h \in S_{K+M,L}} \|h - \sigma_{|I|^2}^2\|_n^2 + \text{pen}(K) \right\} + \frac{C}{Nn} $$

where the penalty function $K \mapsto \text{pen}(K)$ is given by

$$ \text{pen}(K) := \kappa \frac{(K+M)\log^2(N)}{Nn}. \quad (2.19) $$

Considering the adaptive estimator of $\sigma^2$ on the real line $I = \mathbb{R}$ when $N \ge 1$ and $n \to \infty$, we obtain the following result.
**Theorem 2.6.2.** Suppose that $N \ge 1$, $n \to \infty$ and $L = \log(Nn)$, and consider the collection $[\mathbf{B}]$ with

$$K \in \mathcal{K} = \{2^q, q = 0, 1, \dots, q_{\max}\} \subset \mathcal{M} = \{1, \dots, Nn\}.$$

Under Assumption 2.2.1 and for $N$ large enough, the exists a constant $C > 0$ such that,

$$\mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K},L}^2 - \sigma^2 \right\|_{n,N}^2 \right] \le 3 \inf_{K \in \mathcal{K}} \left\{ \inf_{h \in S_{K+M,L}} \left\| h - \sigma^2 \right\|_n^2 + \text{pen}_2(K) \right\} + \frac{C}{\sqrt{Nn}}$$

where the penalty function $\text{pen}_2$ given by

$$\text{pen}_2(K) := \kappa \sqrt{\frac{(K+M) \log^2(Nn)}{Nn}} \qquad (2.20)$$

with $\kappa > 0$ a numerical constant.

We have a penalty term of the same order as the one obtained in Theorem 2.6.1 where $\sigma^2$ is estimated on a compact interval. One can deduce that the adaptive estimator reaches a rate of the same order as the rate of the non-adaptive estimator given in Corollary 2.4.4 for the collection $[\mathbf{B}]$. For the numerical study of the nonparametric estimation of the square of the diffusion coefficient, we rather consider the compact interval $[-\log(Nn), \log(Nn)]$ on which is built the spline basis, and which tends to the real line as $N, n \to \infty$ or $N = 1$ and $n \to \infty$. Then, we use the penalty function $K \mapsto \text{pen}(K)$ given in Equation (2.19).

If we consider the adaptive estimator of the diffusion coefficient, built on a compact interval from a single diffusion path, we obtain below an upper-bound of its risk of estimation.

**Theorem 2.6.3.** Suppose that $N = 1$, $L = \log(n)$, $n \to \infty$ and consider the collection $[\mathbf{B}]$ with

$$K \in \mathcal{K} = \{2^q, q = 0, \dots, q_{\max}\} \subset \mathcal{M} = \{1, \dots, \lfloor \sqrt[5]{n} \rfloor\}.$$

Under Assumption 2.2.1, it holds

$$\mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|}^2 \right\|_{n,1}^2 \right] \le 3 \inf_{K \in \mathcal{K}} \left\{ \inf_{h \in S_{K+M,L}} \left\| h - \sigma_{|I|}^2 \right\|_n^2 + \text{pen}_3(K) \right\} + \frac{C}{n}.$$

where $C > 0$ is a constant depending on $\tau_0$, and

$$\text{pen}_3(K) = \kappa \frac{(K+M) \log(n)}{n}$$

with $\kappa > 0$ a numerical constant.

We deduce from Theorem 2.6.3 that if we assume that $\sigma^2 \in \Sigma_I(\beta, R)$, then the adaptive estimator $\hat{\sigma}_{\hat{K}}^2$ reaches a rate of order $n^{-\beta/(2\beta+1)}$ (up to a log-factor). The result of this theorem is almost a deduction of the result of Theorem 2.6.1, the slight difference being the use, in the proofs, of the local time of the process and the equivalence relation between the pseudo-norm $\|\cdot\|_{n,1}$ with the pseudo-norm $\|\cdot\|_X$ instead of the empirical norm $\|\cdot\|_n$ considered in the proof of Theorem 2.6.1. Since the condition imposed on the dimension of the approximation subspace strongly restricts the model selection, we extend the set of possible values of the dimension to $\mathcal{M} = \{1, \dots, \sqrt{n}/\log(n)\}$ for the numerical study of the estimation of the square $\sigma^2$ of the diffusion coefficient.

## 2.7 Numerical study

This section is devoted to the numerical study on a simulation scheme. Section 2.7.1 focuses on the presentation of the chosen diffusion models. In Section 2.7.2, we describe the scheme for the implementation of the ridge estimators. We mainly focus on the B-spline basis for the numerical study, and in Section 2.7.3, we add a numerical study on the performance of the Hermite-based ridge estimator of $\sigma^2$ on $\mathbb{R}$. Finally, we compare the efficiency of our estimator built on the real line $\mathbb{R}$ from a single path with that of the Nadaraya-Watson estimator proposed in [45].
### 2.7.1 Models and simulations

Recall that the time horizon is $T = 1$ and $X_0 = 0$. Consider the following diffusion models:

Model 1 Ornstein-Uhlenbeck: $b(x) = 1 - x$, $\sigma(x) = 1$

Model 2: $b(x) = 1 - x$, $\sigma(x) = 1 - x^2$

Model 3: $b(x) = 1 - x$, $\sigma(x) = \frac{1}{3+\sin(2\pi x)} + \cos^2(\frac{\pi}{2}x)$

Model 1 is the commonly used Ornstein–Uhlenbeck model, known to be a simple diffusion model satisfying Assumption 2.2.1. Model 2 does not satisfy Assumption 2.2.1. Model 3 satisfies Assumption 2.2.1 with a multimodal diffusion coefficient.

The number $N$ of sample paths takes values in the set $\{1, 10, 100, 1000\}$ where the number $n$ of observations of each diffusion path varies in the set $\{100, 250, 500, 1000\}$. As we work with the spline basis, the dimension $m = K + M$ of the approximation space is chosen such that $M = 3$ and $K$ takes values in $K = \{2^p, p = 0, \dots, 5\}$ so that the subspaces are nested inside each other. We are using $\mathbb{R}$ for the simulation of diffusion paths via the function `sde`. `sim` of `sde` package, (see [62] for more details on the simulation of SDEs).

### 2.7.2 Implementation of the ridge estimators

In this section, we assess the quality of estimation of the adaptive estimator $\hat{\sigma}_{\tilde{m}}^2$ in each of the 3 models through the computation of its risk of estimation. We compare the performance of the adaptive estimator with that of the oracle estimator $\hat{\sigma}_{m^*}^2$ where $m^*$ is given by:

$$m^* := \arg \min_{m \in \mathcal{M}} \|\hat{\sigma}_m^2 - \sigma^2\|_{n,N}^2. \qquad (2.21)$$

For the spline basis, we have $m^* = K^* + M$ with $M = 3$. Finally, we complete the numerical study with a representation of a set of 10 estimators of $\sigma^2$ for each of the 3 models.

We evaluate the MISE of the spline-based adaptive estimators $\hat{\sigma}_{\tilde{K}}^2$ by repeating 100 times the following steps:

1. Simulate observations $D_{N,n}$ and $D_{N',n}$ with $N \in \{1, 10, 100, 1000\}$, $N' = 100$ and $n \in \{100, 250, 1000\}$.

2. For each $K \in \mathcal{K}$, and from $D_{N,n}$, compute estimators $\hat{\sigma}_K^2$ given in Equations (2.7) and (2.8).

3. Select the optimal dimension $\tilde{K} \in \mathcal{K}$ using Equation (2.17) and compute $K^*$ from Equation (2.21).

4. Using $D_{N',n}$, evaluate $\|\hat{\sigma}_{\tilde{K}}^2 - \sigma^2\|_{n,N'}^2$ and $\|\hat{\sigma}_{K^*}^2 - \sigma^2\|_{n,N'}^2$.

We deduce the risks of estimation considering the average values of $\|\hat{\sigma}_{\tilde{m}}^2 - \sigma^2\|_{n,N'}^2$ and $\|\hat{\sigma}_{m^*}^2 - \sigma^2\|_{n,N'}^2$ over the 100 repetitions. Note that we consider in this section, the estimation of $\sigma^2$ on the compact interval $I = [-1, 1]$ and on the real line $\mathbb{R}$. The unknown parameter $\kappa$ of the corresponding penalty functions given in Equations (2.18) and (2.20), are numerically calibrated (details are given in appendix). We choose $\kappa = 4$ for the compact interval $I = [-1, 1]$, and $\kappa = 5$ for the estimation of $\sigma^2$ on the real line $\mathbb{R}$, more precisely, on the interval $[-\log(N), \log(N)]$ which tends to $\mathbb{R}$ as the number $N$ of sample paths tends to infinity.

### 2.7.3 Numerical results

We present in this section the numerical results of the performance of the spline-based adaptive estimators of $\sigma_I^2$ with $I \subseteq \mathbb{R}$ together with the performance of the oracle estimators. We consider the case $I = [-1, 1]$ for the compactly supported diffusion coefficient, and the case $I = \mathbb{R}$.
<table><caption>Table 2.2: Assessment of MISEs (mean and standard deviation between brackets) of both the adaptive estimator $\hat{\sigma}_{\hat{K},L}^2$ and the oracle estimator $\hat{\sigma}_{K*,L}^2$ from diffusion paths of size $n = 100$.</caption><thead><tr><th>Models</th><th>Intervals</th><th>Estimators</th><th>N = 10</th><th>N = 100</th><th>N = 1000</th></tr></thead><tbody><tr><td rowspan="4">Model 1</td><td rowspan="4">[-1, 1]<br>R</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0102 (0.0083)</td><td>0.0009 (0.0009)</td><td>0.0002 (0.0001)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0094 (0.0065)</td><td>0.0009 (0.0009)</td><td>0.0002 (0.0001)</td></tr><tr><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0096 (0.0062)</td><td>0.0009 (0.0008)</td><td>0.0003 (0.0002)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0093 (0.0057)</td><td>0.0009 (0.0008)</td><td>0.0003 (0.0002)</td></tr><tr><td rowspan="4">Model 2</td><td rowspan="4">[-1, 1]<br>R</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0048 (0.0052)</td><td>0.0019 (0.0008)</td><td>0.0005 (0.0002)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0039 (0.0043)</td><td>0.0009 (0.0005)</td><td>0.0005 (0.0002)</td></tr><tr><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0195 (0.0140)</td><td>0.0057 (0.0006)</td><td>0.0012 (0.0002)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0048 (0.0064)</td><td>0.0025 (0.0021)</td><td>0.0010 (0.0003)</td></tr><tr><td rowspan="4">Model 3</td><td rowspan="4">[-1, 1]<br>R</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0521 (0.0191)</td><td>0.0176 (0.0070)</td><td>0.0073 (0.0021)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0260 (0.0081)</td><td>0.0073 (0.0030)</td><td>0.0048 (0.0009)</td></tr><tr><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.1132 (0.0595)</td><td>0.0319 (0.0031)</td><td>0.0179 (0.0054)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0351 (0.0169)</td><td>0.0319 (0.0031)</td><td>0.0116 (0.0051)</td></tr></tbody></table>

<table><caption>Table 2.3: Assessment of MISEs of both the adaptive estimator $\hat{\sigma}_{\hat{K},L}^2$ and the oracle estimator $\hat{\sigma}_{K*,L}^2$ from diffusion paths of size $n = 250$.</caption><thead><tr><th>Models</th><th>Intervals</th><th>Estimators</th><th>N = 10</th><th>N = 100</th><th>N = 1000</th></tr></thead><tbody><tr><td rowspan="4">Model 1</td><td rowspan="4">[-1, 1]<br>R</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0047 (0.0037)</td><td>0.0003 (0.0002)</td><td>0.0001 (0.00003)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0042 (0.0030)</td><td>0.0003 (0.0002)</td><td>0.0001 (0.00003)</td></tr><tr><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0053 (0.0037)</td><td>0.0003 (0.0002)</td><td>0.0001 (0.00004)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0050 (0.0031)</td><td>0.0003 (0.0002)</td><td>0.0001 (0.00004)</td></tr><tr><td rowspan="4">Model 2</td><td rowspan="4">[-1, 1]<br>R</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0027 (0.0019)</td><td>0.0003 (0.0002)</td><td>0.0002 (0.00004)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0018 (0.0019)</td><td>0.0002 (0.0001)</td><td>0.0001 (0.00004)</td></tr><tr><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0091 (0.0077)</td><td>0.0028 (0.0025)</td><td>0.0008 (0.0002)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0020 (0.0023)</td><td>0.0021 (0.0023)</td><td>0.0002 (0.00004)</td></tr><tr><td rowspan="4">Model 3</td><td rowspan="4">[-1, 1]<br>R</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0306 (0.0150)</td><td>0.0058 (0.0012)</td><td>0.0010 (0.0003)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0216 (0.0067)</td><td>0.0023 (0.0020)</td><td>0.0010 (0.0003)</td></tr><tr><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.0560 (0.0313)</td><td>0.0275 (0.0049)</td><td>0.0069 (0.0049)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0261 (0.0127)</td><td>0.0096 (0.0051)</td><td>0.0065 (0.0041)</td></tr></tbody></table>
Tables 2.2 and 2.3 present the numerical results of estimation of $\sigma_I^2$ from simulated data following the steps given in Section 2.7.2.

The results of Table 2.2 and Table 2.3 show that the adapted estimator $\hat{\sigma}_K^2$ is consistent, since its MISE tends to zero as the number $Nn$ of observations is larger. Moreover, note that in most cases, the ridge estimators of the compactly supported diffusion coefficients perform better than those of the non-compactly supported diffusion functions. As expected, we observe that the oracle estimator has generally a better performance compared to the adaptive estimator. Nonetheless, we can remark that the performances are very close in several cases, highlighting the efficiency of the data-driven selection of the dimension.

An additional important remark is the significant influence of the number $n$ of each diffusion path on the performance of $\hat{\sigma}_K^2$ and $\hat{\sigma}_{K^*,L}^2$ (by comparison of Table 2.2 with Table 2.3), which means that estimators built from higher frequency data are more efficient. A similar remark is made for theoretical results obtained in Sections 2.4.2 and 2.4.1.

**Performance of the Hermite-based estimator of the diffusion coefficient** We focus on the estimation of $\sigma^2$ on $\mathbb{R}$ and assess the performance of its Hermite-based estimator (see Section 2.4.2). We present in Table 2.4, the performance of the oracle estimator $\hat{\sigma}_{m^*,L}^2$.

<table><caption>Table 2.4: Assessment of MISEs of the Hermite-based oracle estimator $\hat{\sigma}_{K^*,L}^2$ of the square of the diffusion coefficient.</caption><thead><tr><th>Models</th><th>Intervals</th><th>Estimators</th><th>$N = 10, n = 100$</th><th>$N = 100, n = 100$</th><th>$N = 100, n = 250$</th></tr></thead><tbody><tr><td>Model 1</td><td>$\mathbb{R}$</td><td>$\hat{\sigma}_{K^*,L}^2$</td><td>0.0082 (0.0059)</td><td>0.0015 (0.0008)</td><td>0.0006 (0.0004)</td></tr><tr><td>Model 2</td><td>$\mathbb{R}$</td><td>$\hat{\sigma}_{K^*,L}^2$</td><td>0.0058 (0.0111)</td><td>0.0007 (0.0004)</td><td>0.0003 (0.0002)</td></tr><tr><td>Model 3</td><td>$\mathbb{R}$</td><td>$\hat{\sigma}_{K^*,L}^2$</td><td>0.0188 (0.0151)</td><td>0.0077 (0.0037)</td><td>0.0040 (0.0036)</td></tr></tbody></table>

From the numerical results of Table 2.4, we observe that the Hermite-based estimator of $\sigma^2$ is consistent as the sample size $N$ and the length $n$ paths take larger values.

**Estimation of the diffusion coefficient from one path** Consider ridge estimators of $\sigma_I^2$ with $I = [-1, 1]$. For the case of the adaptive estimators of $\sigma_I^2$, the dimension $\hat{K}$ is selected such that

$$ \hat{K} = \arg \min_{K \in \mathcal{K}} \gamma_n(\hat{\sigma}_K^2) + \text{pen}(K) \quad (2.22) $$

where $\text{pen}(K) = \kappa(K + M) \log(n)/n$ with $\kappa > 0$. We choose the numerical constant $\kappa = 4$ and we derive the numerical performance of the adaptive estimator of $\sigma_I^2$.

Table 2.5 gives the numerical performances of both the adaptive estimator and the oracle estimator of $\sigma_I^2$ on the compact interval $I = [-1, 1]$ and from a single diffusion path. From the obtained results, we see that the estimators are numerically consistent. However, we note that the convergence is slow (increasing $n$ from 100 to 1000), which highlights the significant impact of the number $N$ of paths on the efficiency of the ridge estimator.

**Comparison of the efficiency of the ridge estimator of the diffusion coefficient with its Nadaraya-Watson estimator.** Consider the adaptive estimator $\hat{\sigma}_K^2$ of the square of the diffusion coefficient built on the real line $\mathbb{R}$ from a single diffusion path ($N = 1$), where the dimension $\hat{K}$ is selected using Equation (2.22). For the numerical assessment, we use the interval $I = [-10^6, 10^6]$ to approximate the real line $\mathbb{R}$, and then, use Equation (2.22) for the data-driven selection of the dimension.
<table><caption>Table 2.5: Evaluation of MISEs of adaptive estimators $\hat{\sigma}_{\hat{K},L}^2$ built from a single diffusion path ($N = 1$) for each of the three models.</caption><thead><tr><th>Models</th><th>Intervals</th><th>Estimators</th><th>n = 100</th><th>n = 1000</th></tr></thead><tbody><tr><td rowspan="2">Model 1</td><td rowspan="2">[-1, 1]</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.1751 (0.1921)</td><td>0.0915 (0.1925)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.1563 (0.1776)</td><td>0.0783 (0.1699)</td></tr><tr><td rowspan="2">Model 2</td><td rowspan="2">[-1, 1]</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.1721 (0.3483)</td><td>0.1365 (0.5905)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.0987 (0.1644)</td><td>0.0552 (0.2409)</td></tr><tr><td rowspan="2">Model 3</td><td rowspan="2">[-1, 1]</td><td>σ̂<sup>2</sup><sub>K,L</sub></td><td>0.2184 (0.2780)</td><td>0.2106 (0.5790)</td></tr><tr><td>σ̂<sup>2</sup><sub>K*,L</sub></td><td>0.1263 (0.1486)</td><td>0.0751 (0.1469)</td></tr></tbody></table>

We want to compare the efficiency of $\hat{\sigma}_{\hat{K}}^2$ with that of the Nadaraya-Watson estimator of $\sigma^2$ given from a diffusion path $\bar{X} = (X_{k/n})_{1\le k\le n}$ and for all $x \in \mathbb{R}$ by

$$S_n(x) = \frac{\sum_{k=1}^{n-1} K\left(\frac{X_{k/n} - x}{h_n}\right) [X_{(k+1)/n} - X_{k/n}]^2/n}{\sum_{k=1}^{n} K\left(\frac{X_{k/n} - x}{h_n}\right)}$$

where $K$ is a positive kernel function, and $h_n$ is the bandwidth. Thus, the estimator $S_n(x)$ is consistent under the condition $nh_n^4 \to 0$ as $n$ tends to infinity (see [45]). We use the function `ksdiff()` of the R-package `sde` to compute the Nadaraya-Watson estimator $S_n$.

<table><caption>Table 2.6: This table shows the loss errors of the ridge estimator $\hat{\sigma}_{\hat{K},L}^2$ on $\mathbb{R}$ and the Nadaraya-Watson estimator $S_n$ of $\sigma^2$ built from a diffusion path ($N = 1$) of length $n = 1000$.</caption><thead><tr><th>Models</th><th>Ridge estimator</th><th>Nadaraya-Watson estimator</th></tr></thead><tbody><tr><td>Model 1</td><td>0.0020 (0.0023)</td><td>0.9377 (0.0017)</td></tr><tr><td>Model 2</td><td>0.1323 (0.0794)</td><td>0.5086 (0.0885)</td></tr><tr><td>Model 3</td><td>0.4077 (0.1178)</td><td>1.3175 (0.3039)</td></tr></tbody></table>

We remark from the results of Table 2.6 that our ridge estimator is more efficient. Note that for the kernel estimator $S_n$, the bandwidth is computed using the rule of thumb of Scott (see [80]). The bandwidth is proportional to $n^{-1/(d+4)}$ where $n$ is the number of points, and $d$ is the number of spatial dimensions.

### 2.7.4 Concluding remarks

The results of our numerical study show that our ridge estimators built both on a compact interval and on the real line are consistent as $N$ and $n$ take larger values, or as only $n$ takes larger values when the estimators are built from a single path. These results are in accordance with the theoretical results established in the previous sections. Moreover, as expected, we obtained the consistency of the Hermite-based estimators of $\sigma^2$ on the real line $\mathbb{R}$. Nonetheless, we only focus on the Hermite-based oracle estimator since we did not establish a risk bound of the corresponding adaptive estimator. Finally, we remark that the ridge estimator of $\sigma^2$ built from a single path performs better than its Nadaraya-Watson kernel estimator proposed in [45] and implemented in the R-package `sde`.
<figure><img src="image_10.png" /><figcaption>Figure 2.1: Bundles of 10 estimators $\hat{\sigma}_{K,L}^2$ (in green) of the true diffusion coefficient $\sigma_I^2$ restricted on the compact interval $I = [-1, 1]$ (in red) of each of Models 1, 2, 3 (from top to bottom) using samples of size $N = 1000$ with diffusion paths of length $n = 500$.</figcaption></figure>
## 2.8 Conclusion

In this article, we have proposed ridge-type estimators of the diffusion coefficient on a compact interval from a single diffusion path. We took advantage of the local time of the diffusion process to prove the consistency of non-adaptive estimators of $\sigma^2$ and derive a rate of convergence of the same order than the optimal rate established in [61]. We also propose an estimator of $\sigma^2$ on the real line from a single path. We proved its consistency using the method described in Section 2.3.2, and derive a rate of convergence order $n^{-\beta/(4\beta+1)}$ over a Hölder space for the collection [B]. Then, we extended the study to the estimation of $\sigma^2$ from repeated discrete observations of the diffusion process. We establish rates of convergence of the ridge estimators both on a compact interval and on $\mathbb{R}$. We complete the study proposing adaptive estimators of $\sigma^2$ on a compact interval for $N = 1$ and $N \to \infty$, and on the real line $\mathbb{R}$ for $N \to \infty$.

A perspective on the estimation of the diffusion coefficient could be the establishment of a minimax rate of convergence of the compactly supported (square of the) diffusion coefficient from repeated discrete observations of the diffusion process. The case of the non-compactly supported diffusion coefficient may be a lot more challenging, since the transition density of the diffusion process is no longer lower-bounded. This new fact can lead to different rates of convergence depending on the considered method (see Section 2.4).

## 2.9 Proofs

In this section, we prove our main results of Sections 2.3, 2.4 and 2.6. To simplify our notations, we set $\Delta_n = \Delta(= 1/n)$ and constants are generally denoted by $C > 0$ or $c > 0$ whose values can change from a line to another. Moreover, we use the notation $C_\alpha$ in case we need to specify the dependency of the constant $C$ on a parameter $\alpha$.

### 2.9.1 Technical results

Recall first some useful results on the local time and estimates of the transition density of diffusion processes.

**Lemma 2.9.1.** For all integer $q \ge 1$, there exists $C^* > 0$ depending on $q$ such that for all $0 \le s < t \le 1$,

$$ \mathbb{E} [|X_t - X_s|^{2q}] \le C^*(t-s)^q. $$

The proof of Lemma 2.9.1 is provided in [28].

**Proposition 2.9.2.** Under Assumptions 2.2.1, there exist constants $c_\sigma > 1$, $C > 1$ such that for all $t \in (0, 1]$, $x \in \mathbb{R}$,

$$ \frac{1}{C\sqrt{t}} \exp\left(-c_{\sigma}\frac{x^2}{t}\right) \le p_X(t, x) \le \frac{C}{\sqrt{t}} \exp\left(-\frac{x^2}{c_{\sigma}t}\right). $$

The proof of Proposition 2.9.2 is provided in [55], Proposition 1.2.

**Proposition 2.9.3.** Let $h$ be a $L_0$-Lipschitz function. Then there exists $\tilde{h} \in S_{K_N, M}$, such that

$$ |\tilde{h}(x) - h(x)| \le C \frac{\log(N)}{K_N}, \quad \forall x \in (-\log(N), \log(N)), $$

where $C > 0$ depends on $L_0$, and $M$.

The proof of Proposition 2.9.3 is provided in [28]. The finite-dimensional vector space $S_{K_N, M} = S_{K_N+M}$ is introduced in Section 2.2.

**Lemma 2.9.4.** Under Assumption 2.2.1, there exist $C_1, C_2 > 0$ such that for all $A > 0$,

$$ \sup_{t \in [0,1]} \mathbb{P}(|X_t| \ge A) \le \frac{C_1}{A} \exp(-C_2 A^2). $$
The proof of Lemma 2.9.4 is provided in [28], Lemma 7.3.

**Lemma 2.9.5.** Under Assumption 2.2.1, the following holds:

$$ \forall x \in \mathbb{R}, \quad \mathcal{L}^x = \mathcal{L}^{x-} \quad \text{a.s.} $$

$$ \text{where } \mathcal{L}^{x-} = \lim_{\epsilon \to 0} \mathcal{L}^{x-\epsilon}. $$

The result of Lemma 2.9.5 justifies the definition of the local time $\mathcal{L}^x$, for $x \in \mathbb{R}$, given in Equation (2.10). The proof of Lemma 2.9.5 is provided in appendix.

## 2.9.2 Proofs of Section 2.3

### Proof of Theorem 2.3.2

Let $\Omega_{n,m}$ be a random event given by

$$ \Omega_{n,m} := \bigcap_{g \in S_m \setminus \{0\}} \left\{ \left| \frac{\|g\|_{n,1}^2}{\|g\|_X^2} - 1 \right| \le \frac{1}{2} \right\}. $$

On the event $\Omega_{n,m}$, the two pseudo-norms $\|\cdot\|_{n,1}$ and $\|\cdot\|_X$ are equivalent with $(1/2)\|g\|_X^2 \le \|g\|_{n,1}^2 \le (3/2)\|g\|_X^2$ for each function $g \in S_m \setminus \{0\}$. The proof of Theorem 2.3.2 relies on the following lemma.

**Lemma 2.9.6.** Let $\gamma > 1$ be a real number. Under Assumption 2.2.1, the following holds

$$ \mathbb{P}(\Omega_{n,m}^c) \le C \frac{m^{\alpha\gamma}}{n^{\gamma/2}}, $$

where $C > 0$ is a constant depending on $\gamma$, with $\alpha = 2$ for the collection $[\mathbf{B}]$, and $\alpha = (r + r')/2$ for the collection $[\mathbf{CS}-\mathbf{OB}]$.

The parameter $\gamma > 1$ has to be chosen appropriately so that we obtain a variance term of the risk of the estimator $\hat{\sigma}_m^2$ that is negligible with respect to $m^{\alpha'}/n$ as $n$ tends to infinity (see Theorem 2.3.2 and Corollary 2.3.3). The proof of Lemma 2.9.6 is provided in appendix.

**Proof of Theorem 2.3.2.** Recall that since $N = 1$, $\zeta_{k\Delta}^1 = \zeta_{k\Delta}^{1,1} + \zeta_{k\Delta}^{1,2} + \zeta_{k\Delta}^{1,3}$ is the error term of the regression model, with:

$$ \zeta_{k\Delta}^{1,1} = \frac{1}{\Delta} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s^1 \right)^2 - \int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_s^1) ds \right], \quad (2.23) $$

$$ \zeta_{k\Delta}^{1,2} = \frac{2}{\Delta} \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s)\sigma'(X_s^1)\sigma^2(X_s^1) dW_s^1, \quad (2.24) $$

$$ \zeta_{k\Delta}^{1,3} = 2b(X_{k\Delta}^1) \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s^1. \quad (2.25) $$

Besides, $R_{k\Delta}^1 = R_{k\Delta}^{1,1} + R_{k\Delta}^{1,2} + R_{k\Delta}^{1,3}$, with:

$$ R_{k\Delta}^{1,1} = \frac{1}{\Delta} \left( \int_{k\Delta}^{(k+1)\Delta} b(X_s^1) ds \right)^2, \quad R_{k\Delta}^{1,2} = \frac{1}{\Delta} \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s) \Phi(X_s^1) ds \quad (2.26) $$

$$ R_{k\Delta}^{1,3} = \frac{2}{\Delta} \left( \int_{k\Delta}^{(k+1)\Delta} (b(X_s^1) - b(X_{k\Delta}^1)) ds \right) \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s^1 \right) \quad (2.27) $$

where

$$ \Phi := 2b\sigma'\sigma + [\sigma''\sigma + (\sigma')^2]\sigma^2. \quad (2.28) $$
By definition of the projection estimator $\hat{\sigma}_m^2$ for each $m \in \mathcal{M}$ (see Equation (2.7)), for all $h \in S_{m,L}$, we have:

$$ \gamma_{n,1}(\hat{\sigma}_m^2) - \gamma_{n,1}(\sigma_{|I}^2) \leq \gamma_{n,1}(h) - \gamma_{n,1}(\sigma_{|I}^2). \quad (2.29) $$

Furthermore, for all $h \in S_{m,L}$,

$$ \gamma_{n,1}(h) - \gamma_{n,1}(\sigma_{|I}^2) = \|\sigma_{|I}^2 - h\|_{n,1}^2 + 2\nu_1(\sigma_{|I}^2 - h) + 2\nu_2(\sigma_{|I}^2 - h) + 2\nu_3(\sigma_{|I}^2 - h) + 2\mu(\sigma_{|I}^2 - h), $$

where,

$$ \nu_i(h) = \frac{1}{n} \sum_{k=0}^{n-1} h(X_{k\Delta}^1) \zeta_{k\Delta}^{1,i}, \quad i \in \{1, 2, 3\}, \quad \mu(h) = \frac{1}{n} \sum_{k=0}^{n-1} h(X_{k\Delta}^1) R_{k\Delta}^1, \quad (2.30) $$

and $\zeta_{k\Delta}^{1,1}$, $\zeta_{k\Delta}^{1,2}$, $\zeta_{k\Delta}^{1,3}$ are given in Equations (2.23), (2.24), (2.25), and finally, $R_{k\Delta}^1 = R_{k\Delta}^{1,1} + R_{k\Delta}^{1,2}$ given in Equations (2.26) and (2.27). Then, for all $m \in \mathcal{M}$, and for all $h \in S_{m,L}$, we obtain from Equation (2.29) that

$$ \|\hat{\sigma}_m^2 - \sigma_{|I}^2\|_{n,1}^2 \le \|h - \sigma_{|I}^2\|_{n,1}^2 + 2\nu(\hat{\sigma}_m^2 - h) + 2\mu(\hat{\sigma}_m^2 - h), \quad \text{with } \nu = \nu_1 + \nu_2 + \nu_3. $$

Then, it comes,

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I}^2 \right\|_{n,1}^2 \right] \le \inf_{h \in S_{m,L}} \left\| h - \sigma_{|I}^2 \right\|_n^2 + 2\mathbb{E} \left[ \nu (\hat{\sigma}_m^2 - h) \right] + 2\mathbb{E} \left[ \mu (\hat{\sigma}_m^2 - h) \right]. \quad (2.31) $$

Besides, for any $a, d > 0$, using the inequality $xy \le \eta x^2 + y^2/\eta$ with $\eta = a, d$, we have,

$$ \left\{ \begin{array}{l} 2\nu (\hat{\sigma}_m^2 - h) \le \frac{2}{a} \|\hat{\sigma}_m^2 - \sigma_{|I}^2\|_X^2 + \frac{2}{a} \|h - \sigma_{|I}^2\|_X^2 + a \sup_{h \in S_m, \|h\|_X=1} \nu^2(h), \\ \\ 2\mu (\hat{\sigma}_m^2 - h) \le \frac{2}{d} \|\hat{\sigma}_m^2 - \sigma_{|I}^2\|_{n,1}^2 + \frac{2}{d} \|h - \sigma_{|I}^2\|_{n,1}^2 + \frac{d}{n} \sum_{k=1}^n (R_{k\Delta}^1)^2. \end{array} \right. \quad (2.32) $$

**Upper bound of** $\frac{1}{n} \sum_{k=1}^n (R_{k\Delta}^1)^2$

We have:

$$ \forall k \in [1, n], R_{k\Delta}^1 = R_{k\Delta}^{1,1} + R_{k\Delta}^{1,2} + R_{k\Delta}^{1,3} \text{ with,} $$

$$ R_{k\Delta}^{1,1} = \frac{1}{\Delta} \left( \int_{k\Delta}^{(k+1)\Delta} b(X_s^1) ds \right)^2, \quad R_{k\Delta}^{1,2} = \frac{1}{\Delta} \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s) \Phi(X_s^1) ds $$

$$ R_{k\Delta}^{1,3} = \frac{2}{\Delta} \left( \int_{k\Delta}^{(k+1)\Delta} (b(X_s^1) - b(X_{k\Delta}^1)) ds \right) \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s^1 \right). $$

For all $k \in [1, n]$, using the Cauchy-Schwarz inequality and Lemma (2.2.2),

$$ \mathbb{E} \left[ \left| R_{k\Delta}^{1,1} \right|^2 \right] \le \mathbb{E} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} b^2(X_{k\Delta}^1) ds \right)^2 \right] \le \Delta \mathbb{E} \left[ \int_{k\Delta}^{(k+1)\Delta} b^4(X_{k\Delta}^1) ds \right] \le C \Delta^2. $$

Consider now the term $R_{k\Delta}^{1,2}$. From Equation (2.28), we have $\Phi = 2b\sigma'\sigma + [\sigma''\sigma + (\sigma')^2]\sigma^2$ and according to Assumption 2.2.1, there exists a constant $C > 0$ depending on $\sigma_1$ and $\alpha$ such that

$$ |\Phi(X_s^1)| \le C [(2 + |X_s^1|)(1 + |X_s^1|^\alpha) + (1 + |X_s^1|^\alpha)^2]. $$

Then, from Lemma (2.2.2) and for all $s \in (0, 1]$,

$$ \mathbb{E}[\Phi^2(X_s^1)] \le C \sup_{s \in (0,1]} \mathbb{E} \left[ (2 + |X_s^1|^2)(1 + |X_s^1|^\alpha)^2 + (1 + |X_s^1|^\alpha)^4 \right] < \infty $$
and

$$ \mathbb{E} \left[ \left| R_{k\Delta}^{1,2} \right|^2 \right] \le \frac{1}{\Delta^2} \int_{k\Delta}^{(k+1)\Delta} \left( (k+1)\Delta - s \right)^2 ds \int_{k\Delta}^{(k+1)\Delta} \mathbb{E} \left[ \Phi^2(X_s^1) \right] ds \le C\Delta^2 $$

Finally, under Assumption 2.2.1, from Lemma (2.2.2) and using the Cauchy-Schwarz inequality, we have

$$ \begin{aligned} \mathbb{E} \left[ \left| R_{k\Delta}^{1,3} \right|^2 \right] & \le \frac{4}{\Delta^2} \mathbb{E} \left[ \Delta \int_{k\Delta}^{(k+1)\Delta} L_0^2 |X_s^1 - X_{k\Delta}^1|^2 ds \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s \right)^2 \right] \\ & \le \frac{4}{\Delta} \sqrt{\mathbb{E} \left[ L_0^4 \Delta \int_{k\Delta}^{(k+1)\Delta} |X_s^1 - X_{k\Delta}^1|^4 ds \right]} \mathbb{E} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s \right)^4 \right] \\ & \le C\Delta^2. \end{aligned} $$

As a result, there exists a constant $C > 0$ such that,

$$ \mathbb{E} \left[ \frac{1}{n} \sum_{k=1}^{n} (R_{k\Delta}^{1})^2 \right] \le C\Delta^2. \quad (2.33) $$

We set $a = d = 8$ and considering the event $\Omega_{n,m}$ on which the empirical norms $\|\cdot\|_X$ and $\|\cdot\|_{n,1}$ are equivalent, we deduce from Equations (2.31), (2.32) and (2.33) that,

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|}^2 \right\|_{n,1}^2 \mathbb{1}_{\Omega_{n,m}} \right] \le 3 \inf_{h \in \mathcal{S}_m} \left\| h - \sigma_{|I|}^2 \right\|_n^2 + C \mathbb{E} \left( \sup_{h \in \mathcal{S}_m, \|h\|_X = 1} \nu^2(h) \right) + C\Delta^2 \quad (2.34) $$

where $C > 0$ is a constant depending on $\sigma_1$.

$$ \textbf{Upper bound of } \mathbb{E} \left( \sup_{h \in \mathcal{S}_m, \|h\|_X = 1} \nu^2(h) \right) $$

For all $h = \sum_{\ell=0}^{m-1} a_\ell \phi_\ell \in \mathcal{S}_m$ such that $\|h\|_X^2 = 1$, we have $\|h\|^2 \le \frac{1}{\tau_0}$ (see Equation (2.11)) and the coordinate vector $\mathbf{a} = (a_{-M}, \dots, a_{K-1})$ satisfies:

* $\|\mathbf{a}\|_2^2 \le Cm$ ($m = K + M$) for the collection [B] (see [30], Lemma 2.6)

* $\|\mathbf{a}\|_2^2 \le 1/\tau_0$ for the collection [CS-OB] since $\|h\|^2 = \|\mathbf{a}\|_2^2$.

Furthermore, using the Cauchy-Schwarz inequality, we have:

$$ \nu^2(h) = \left( \sum_{\ell=0}^{m-1} a_\ell \nu(\phi_\ell) \right)^2 \le \|\mathbf{a}\|_2^2 \sum_{\ell=0}^{m-1} \nu^2(\phi_\ell). $$

Thus, since $\nu = \nu_1 + \nu_2 + \nu_3$, for all $\ell \in [-M, K-1]$ and for all $i \in \{1, 2, 3\}$,

$$ \mathbb{E} \left[ \nu_i^2 (\phi_\ell) \right] = \frac{1}{n^2} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \phi_\ell (X_{k\Delta}^1) \zeta_{k\Delta}^{1,i} \right)^2 \right]. $$

1. Case $i = 1$

Recall that $\zeta_{k\Delta}^{1,1} = \frac{1}{\Delta} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s \right)^2 - \int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_s^1) ds \right]$ where $W = W^1$. We fix a initial time $s \in [0, 1)$ and set $M_t^s = \int_s^t \sigma(X_u^1) dW_u$, $\forall t \ge s$. $(M_t^s)_{t \ge s}$ is a martingale and for all $t \in [s, 1]$, we have:

$$ \langle M^s, M^s \rangle_t = \int_s^t \sigma^2(X_u^1) du. $$
Then, $\zeta_{k\Delta}^{1,1} = \frac{1}{\Delta} (M_{(k+1)\Delta}^{k\Delta})^2 - \langle M^{k\Delta}, M^{k\Delta} \rangle_{(k+1)\Delta}$ is also a $\mathcal{F}_{k\Delta}$-martingale, and, using the Burkholder-Davis-Gundy inequality, we obtain for all $k \in [0, n-1]$,

$$ \mathbb{E}[\zeta_{k\Delta}^{1,1}|\mathcal{F}_{k\Delta}] = 0, \quad \mathbb{E}\left[(\zeta_{k\Delta}^{1,1})^2|\mathcal{F}_{k\Delta}\right] \le \frac{C}{\Delta^2}\mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_u^1)du\right)^2\right] \le C\sigma_1^4. \quad (2.35) $$

Then, using Equation (2.35) we have:

$$ \begin{aligned} \mathbb{E}[\nu_1^2(\phi_\ell)] &= \frac{1}{n^2}\mathbb{E}\left[\sum_{k=0}^{n-1} \phi_\ell^2(X_{k\Delta}^1) (\zeta_{k\Delta}^{1,1})^2\right] = \frac{1}{n^2}\mathbb{E}\left[\sum_{k=0}^{n-1} \phi_\ell^2(X_{k\Delta}^1) \mathbb{E}\left[(\zeta_{k\Delta}^{1,1})^2|\mathcal{F}_{k\Delta}\right]\right] \\ &\le \frac{C\sigma_1^4}{n^2}\mathbb{E}\left[\sum_{k=0}^{n-1} \phi_\ell^2(X_{k\Delta}^1)\right] \end{aligned} $$

and,

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_1^2(\phi_\ell)] \le \frac{C\sigma_1^4}{n^2} \mathbb{E}\left[\sum_{k=0}^{n-1} \sum_{\ell=0}^{m-1} \phi_\ell^2(X_{k\Delta}^1)\right]. $$

One has:

$$ \begin{cases} \sum_{\ell=-M}^{K-1} B_\ell^2(X_{\eta(s)}^1) \le 1 & \text{for the Collection } [\mathbf{B}] \ (m = K + M), \\ \sum_{\ell=0}^{m-1} \phi_\ell^2(X_{\eta(s)}^1) \le Cm^r & \text{for the collection } [\mathbf{CS} - \mathbf{OB}]. \end{cases} $$

Thus, it follows that

* $\sum_{\ell=-M}^{K-1} \mathbb{E}[\nu_1^2(B_\ell)] \le C/n$ for the Spline basis,

* $\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_1^2(\phi_\ell)] \le Cm^r/n$ for the collection $[\mathbf{CS}-\mathbf{OB}]$,

and,

$$ \mathbb{E}\left(\sup_{h \in S_m, \|h\|_X^2=1} \nu_1^2(h)\right) \le C \frac{m^{\alpha'}}{n} \quad (2.36) $$

where $C > 0$ is a constant depending on $\sigma_1$ and the basis, with $\alpha' = 1$ for the collection $[\mathbf{B}]$, and $\alpha' = r$ for the collection $[\mathbf{CS}-\mathbf{OB}]$.

## 2. Case $i = 2$

We have $\zeta_{k\Delta}^{1,2} = \frac{2}{\Delta} \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s) \sigma'(X_s^1) \sigma^2(X_s^1) dW_s$ and,

$$ \begin{aligned} \mathbb{E}[\nu_2^2(\phi_\ell)] &= 4\mathbb{E}\left[\left(\sum_{k=0}^{n-1} \phi_\ell(X_{k\Delta}^1) \int_{k\Delta}^{(k+1)\Delta} (k+1)\Delta - s \right) \sigma'(X_s^1) \sigma^2(X_s^1) dW_s\right)^2\right] \\ &= 4\mathbb{E}\left[\left(\int_0^1 \phi_\ell(X_{\eta(s)}^1) (\eta(s) + \Delta - s) \sigma'(X_s^1) \sigma^2(X_s^1) dW_s\right)^2\right] \\ &\le C\sigma_1^4\Delta^2\mathbb{E}\left[\int_0^1 \phi_\ell^2(X_{\eta(s)}^1) ds\right] \end{aligned} $$

where $C > 0$ is a constant. We deduce for both the spline basis and any orthonormal basis that there exists a constant $C > 0$ depending on $\sigma_1$ such that:

$$ \mathbb{E}\left(\sup_{h \in S_m, \|h\|_X^2=1} \nu_2^2(h)\right) \le C \frac{m^{\alpha'}}{n^2}. \quad (2.37) $$
3. Case $i = 3$

We have $\zeta_{k\Delta}^{1,3} = 2b(X_{k\Delta}^1) \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s$ and,

$$
\begin{aligned}
\mathbb{E}[\nu_3^2(\phi_\ell)] &= \frac{4}{n^2} \mathbb{E} \left[ \left( \int_0^1 \phi_\ell(X_{\eta(s)}^1) b(X_{\eta(s)}^1) \sigma(X_s^1) dW_s \right)^2 \right] \\
&\leq \frac{4\sigma_1^2}{n^2} \mathbb{E} \left[ \int_0^1 \phi_\ell^2(X_{\eta(s)}^1) b^2(X_{\eta(s)}^1) ds \right]
\end{aligned}
$$

Since for all $x \in \mathbb{R}$, $b^2(x) \le C_0(1+x^2)$ and $\sup_{t \in [0,1]} \mathbb{E}(|X_t|^2) < \infty$, there exists a constant $C > 0$ depending on $\sigma_1$ such that:

$$
\mathbb{E} \left( \sup_{h \in S_m, \|h\|_X^2=1} \nu_3^2(h) \right) \le C \frac{m^{\alpha'}}{n^2}. \qquad (2.38)
$$

We finally obtain from Equations (2.36), (2.37) and (2.38) that there exists a constant $C > 0$ depending on $\sigma_1$ such that:

$$
\mathbb{E} \left( \sup_{h \in S_m, \|h\|_X^2=1} \nu^2(h) \right) \le C \frac{m^{\alpha'}}{n}. \qquad (2.39)
$$

We deduce from Equations (2.34) and (2.39) that there exists a constant $C > 0$ depending on $\sigma_1$ such that,

$$
\mathbb{E} \left[ \| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \|_{n,1}^2 \mathbf{1}_{\Omega_{n,m}} \right] \le 3 \inf_{h \in S_{m,L}} \|\sigma_{|I|^2}^2 - h\|_n^2 + C \left( \frac{m^{\alpha'}}{n} + \Delta^2 \right).
$$

For $n$ large enough, we have $\|\hat{\sigma}_m^2 - \sigma_{|I|^2}^2\|_{\infty}^2 \le 2mL$ since $\|\hat{\sigma}_m^2\|_{\infty} \le \sqrt{mL}$. Then, from Lemma 2.9.6 and for all $m \in \mathcal{M}$, there exists a constant $C > 0$ depending on $\sigma_1$ such that

$$
\begin{aligned}
\mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_{n,1}^2 \right] &= \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_{n,1}^2 \mathbf{1}_{\Omega_{n,m}} \right] + \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_{n,1}^2 \mathbf{1}_{\Omega_{n,m}^c} \right] \\
&\le \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_{n,1}^2 \mathbf{1}_{\Omega_{n,m}} \right] + 2mLP \left( \Omega_{n,m}^c \right) \\
&\le 3 \inf_{h \in S_{m,L}} \|\sigma_{|I|^2}^2 - h\|_n^2 + C \left( \frac{m^{\alpha'}}{n} + \frac{m^{\alpha\gamma+1}L}{n^{\gamma/2}} + \Delta^2 \right).
\end{aligned}
$$

Since the pseudo-norms $\|\cdot\|_{n,1}$ and $\|\cdot\|_X$ are equivalent on the event $\Omega_{n,m}$, then, using Lemma 2.9.6, there exists a constant $C > 0$ depending on $\sigma_1$ such that

$$
\begin{aligned}
\mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_X^2 \right] &= \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_X^2 \mathbf{1}_{\Omega_{n,m}} \right] + \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_X^2 \mathbf{1}_{\Omega_{n,m}^c} \right] \\
&\le 8 \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_{n,1}^2 \right] + 10 \inf_{h \in S_m} \|\sigma_{|I|^2}^2 - h\|_n^2 + 2mLP \left( \Omega_{n,m}^c \right) \\
&\le 34 \inf_{h \in S_{m,L}} \|h - \sigma_{|I|^2}^2\|_n^2 + C \left( \frac{m^{\alpha'}}{n} + \frac{m^{\alpha\gamma+1}L}{n^{\gamma/2}} + \Delta^2 \right).
\end{aligned}
$$

Finally, since the estimator $\hat{\sigma}_m^2$ is built from a diffusion path $\bar{X}^1$ independent of the diffusion process $X$, and from Equations (2.11) and (2.9), the pseudo-norm $\|\cdot\|_X$ depending on the process $X$ and the empirical norm $\|\cdot\|_n$ are equivalent ($\forall h \in \mathbb{L}^2(I)$, $\|h\|_n^2 \le (\tau_1/\tau_0)\mathbb{E}[\|h\|_X^2]$), there exists a constant $C > 0$ depending on $\sigma_1$, $\tau_0$ and $\tau_1$ such that

$$
\mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2}^2 \right\|_n^2 \right] \le \frac{34\tau_1}{\tau_0} \inf_{h \in S_{m,L}} \|h - \sigma_{|I|^2}^2\|_n^2 + C \left( \frac{m^{\alpha'}}{n} + \frac{m^{\alpha\gamma+1}L}{n^{\gamma/2}} + \Delta^2 \right).
$$

□
**Proof of Theorem 2.3.4**

*Proof.* Since $L = \log^2(n)$. From Equation (2.31) (Proof of Theorem 2.3.2), for all $h \in S_{m,L}$,

$$ \mathbb{E} \left[ \left\| (\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2) \right\|_{n,1}^2 \right] \le \inf_{h \in S_{m,L}} \left\| h - \sigma_{\mathcal{I}}^2 \right\|_n^2 + 2 \sum_{i=1}^3 \mathbb{E} [\nu_i (\hat{\sigma}_m^2 - h)] + 2 \mathbb{E} [\mu (\hat{\sigma}_m^2 - h)] \quad (2.40) $$

where $\nu_i$, $i = 1, 2, 3$ and $\mu$ are given in Equation (2.30). For all $i \in \{1, 2, 3\}$ and for all $h \in S_{m,L}$, one has

$$ \mathbb{E} [\nu_i (\hat{\sigma}_{m,L}^2 - h)] \le \sqrt{2m \log^2(n)} \sqrt{\sum_{\ell=0}^{m-1} \mathbb{E} [\nu_i^2(\phi_\ell)]}. \quad (2.41) $$

1. Upper bound of $\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_1^2(\phi_\ell)]$

According to Equation (2.30), we have

$$ \forall \ell \in [0, m-1], \nu_1(\phi_\ell) = \frac{1}{n} \sum_{k=0}^{n-1} \phi_\ell(X_{k\Delta}^1) \zeta_{k\Delta}^{1,1} $$

where $\zeta_{k\Delta}^{1,1} = \frac{1}{\Delta} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s \right)^2 - \int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_s^1) ds \right]$ is a martingale satisfying

$$ \mathbb{E}[\zeta_{k\Delta}^{1,1} | \mathcal{F}_{k\Delta}] = 0 \quad \text{and} \quad \mathbb{E}\left[(\zeta_{k\Delta}^{1,1})^2 | \mathcal{F}_{k\Delta}\right] \le \frac{1}{\Delta^2} \mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_s^1) ds\right)^2\right] \le C\sigma_1^4 $$

with $C > 0$ a constant, $W = W^1$ and $(\mathcal{F}_t)_{t \ge 0}$ the natural filtration of the martingale $(M_t)_{t \in [0,1]}$ given for all $t \in [0, 1]$ by $M_t = \int_0^t \sigma(X_s^1) dW_s$. We derive that

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_1^2(\phi_\ell)] = \frac{1}{n^2} \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \phi_\ell(X_{k\Delta}^1) \zeta_{k\Delta}^{1,1} \right)^2 \right] = \frac{1}{n^2} \mathbb{E} \left[ \sum_{k=0}^{n-1} \sum_{\ell=0}^{m-1} \phi_\ell^2(X_{k\Delta}^1) (\zeta_{k\Delta}^{1,1})^2 \right] $$

since for all integers $k, k'$ such that $k > k' \ge 0$, we have

$$ \mathbb{E}[\phi_{\ell}(X_{k\Delta}^{1})\zeta_{k\Delta}^{1,1}\phi_{\ell'}(X_{k'\Delta}^{1})\zeta_{k'\Delta}^{1,1}|\mathcal{F}_{k\Delta}] = \phi_{\ell}(X_{k\Delta}^{1})\zeta_{k'\Delta}^{1,1}\phi_{\ell'}(X_{k'\Delta}^{1})\mathbb{E}[\zeta_{k\Delta}^{1,1}|\mathcal{F}_{k\Delta}] = 0. $$

For each $k \in [0, n-1]$, we have

$$ \left\{ \begin{array}{l} \sum_{\ell=0}^{m-1} \phi_{\ell}(X_{k\Delta}^{1}) = \sum_{\ell=-M}^{K-1} B_{\ell}(X_{k\Delta}^{1}) = 1 \quad \text{for the spline basis} \\ \\ \sum_{\ell=0}^{m-1} \phi_{\ell}(X_{k\Delta}^{1}) \le Cm \quad \text{For an orthonormal basis with } C = \max_{0 \le \ell \le m-1} \|\phi_{\ell}\|_{\infty}. \end{array} \right. $$

Finally, there exists a constant $C > 0$ such that

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_1^2(\phi_\ell)] \le \begin{cases} \frac{C}{n} & \text{for the spline basis} \\ C \frac{m}{n} & \text{for an orthonormal basis.} \end{cases} $$

2. Upper bound of $\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_2^2(\phi_\ell)]$

For all $k \in [0, n-1]$ and for all $s \in [0, 1]$, set $\eta(s) = k\Delta$ if $s \in [k\Delta, (k+1)\Delta)$. We have:

$$ \begin{aligned} \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_2^2(\phi_\ell)] &= 4 \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \int_{k\Delta}^{(k+1)\Delta} \phi_\ell(X_{k\Delta}^1) ((k+1)\Delta - s) \sigma'(X_s^1) \sigma^2(X_s^1) dW_s \right)^2 \right] \\ &= 4 \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \int_0^1 \phi_\ell(X_{\eta(s)}^1) (\eta(s) + \Delta - s) \sigma'(X_s^1) \sigma^2(X_s^1) dW_s \right)^2 \right]. \end{aligned} $$
We conclude that

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_2^2(\phi_\ell)] \le \begin{cases} \frac{C}{n^2} & \text{for the spline basis} \\ C \frac{m}{n^2} & \text{for an orthonormal basis.} \end{cases} $$

where the constant $C > 0$ depends on the diffusion coefficient and the upper bound of the basis functions.

3. Upper bound of $\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_2^3(\phi_\ell)]$

We have:

$$ \begin{aligned} \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_3^2(\phi_\ell)] &= \frac{4}{n^2} \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \int_{k\Delta}^{(k+1)\Delta} \phi_\ell(X_{k\Delta}^1)b(X_{k\Delta}^1)\sigma(X_s^1)dW_s \right)^2 \right] \\ &= \frac{4}{n^2} \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \int_0^1 \phi_\ell(X_{\eta(s)}^1)b(X_{\eta(s)}^1)\sigma(X_s^1)dW_s \right)^2 \right] \\ &\le \frac{4}{n^2} \mathbb{E} \left[ \int_0^1 \sum_{\ell=0}^{m-1} \phi_\ell^2(X_{\eta(s)}^1)b^2(X_{\eta(s)}^1)\sigma^2(X_s^1)ds \right]. \end{aligned} $$

Since for all $x \in \mathbb{R}$, $b(x) \le C_0(1+x^2)$ and $\sup_{t \in [0,1]} \mathbb{E}(|X_t|^4) < \infty$, there exists a constant $C > 0$ depending on the diffusion coefficient such that

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_3^2(\phi_\ell)] \le \begin{cases} \frac{C}{n^2} & \text{for the spline basis} \\ C \frac{m}{n^2} & \text{for an orthonormal basis.} \end{cases} $$

We finally deduce that from Equations (2.40) and (2.41) that for all $h \in S_{m,L}$,

$$ \left\{ \begin{aligned} \mathbb{E} \left[ \| (\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2) \|_{n,1}^2 \right] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m \log^2(n)}{n}} + 2 \mathbb{E} \left[ \mu(\hat{\sigma}_{m,L}^2 - h) \right] \quad [\mathbf{B}] \\ \mathbb{E} \left[ \| (\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2) \|_{n,1}^2 \right] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m^2 \log^2(n)}{n}} + 2 \mathbb{E} \left[ \mu(\hat{\sigma}_{m,L}^2 - h) \right] \quad [\mathbf{OB}] \end{aligned} \right. \quad (2.42) $$

where $C > 0$ is a constant. It remains to obtain an upper bound of the term $\mu(\hat{\sigma}_{m,L}^2 - h)$. For all $a > 0$ and for all $h \in S_{m,L}$,

$$ \begin{aligned} 2\mu (\hat{\sigma}_{m,L}^2 - h) &\le \frac{2}{a} \|\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2\|_{n,1}^2 + \frac{2}{a} \|h - \sigma_{\mathcal{I}}^2\|_{n,1}^2 + \frac{a}{n} \sum_{k=0}^{n-1} (R_{k\Delta}^1)^2 \\ 2\mathbb{E} [\mu (\hat{\sigma}_{m,L}^2 - h)] &\le \frac{2}{a} \mathbb{E} \|\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2\|_{n,1}^2 + \frac{2}{a} \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + \frac{a}{n} \sum_{k=0}^{n-1} \mathbb{E} [(R_{k\Delta}^1)^2]. \end{aligned} $$

Using Equations (2.33), (2.42) and setting $a = 4$, we deduce that there exists constant $C > 0$ depending on $\sigma_1$ such that,

$$ \left\{ \begin{aligned} \mathbb{E} \left[ \| \hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2 \|_{n,1}^2 \right] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m \log^2(n)}{n}} \quad [\mathbf{B}] \\ \mathbb{E} \left[ \| \hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2 \|_{n,1}^2 \right] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m^2 \log^2(n)}{n}} \quad [\mathbf{OB}]. \end{aligned} \right. \quad (2.43) $$
For the case of bases of compactly supported functions, we proceed to the dilation of the bases so that $\sigma^2$ can be estimated on the compact interval $[-\log(n), \log(n)]$. Then, we use the following decomposition:

$$ 
\begin{aligned}
\mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma^2\|_{n,1}^2] &\le \mathbb{E} [\|(\hat{\sigma}_{m,L}^2 - \sigma^2)\mathbf{1}_{[-\log(n),\log(n)]}\|_{n,1}^2] + \mathbb{E} [\|(\hat{\sigma}_{m,L}^2 - \sigma^2)\mathbf{1}_{[-\log(n),\log(n)]^c}\|_{n,1}^2] \\
&\le \mathbb{E} [\|(\hat{\sigma}_{m,L}^2 - \sigma^2)\mathbf{1}_{[-\log(n),\log(n)]}\|_{n,1}^2] + 4L \sup_{t \in [0,1]} \mathbb{P}(|X_t| > \log(n)),
\end{aligned}
$$

and we deduce from Equation (2.43) that

$$ 
\left\{
\begin{aligned}
\mathbb{E} [\|(\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2)\mathbf{1}_{[-\log(n),\log(n)]}\|_{n,1}^2] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m \log^2(n)}{n}} \quad [\mathbf{B}] \\
\mathbb{E} [\|(\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2)\mathbf{1}_{[-\log(n),\log(n)]}\|_{n,1}^2] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m^2 \log^2(n)}{n}} \quad [\mathbf{OB}],
\end{aligned}
\right.
$$

and

$$ 
\left\{
\begin{aligned}
\mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2\|_{n,1}^2] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m \log^2(n)}{n}} + 2 \log^2(n) \sup_{t \in (0,1]} \mathbb{P}(|X_t| > A_n) \quad [\mathbf{B}] \\
\mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2\|_{n,1}^2] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m^2 \log^2(n)}{n}} + 2 \log^2(n) \sup_{t \in (0,1]} \mathbb{P}(|X_t| > A_n) \quad [\mathbf{OB}].
\end{aligned}
\right.
\quad (2.44) $$

From Proposition 2.9.3, $\sup_{t \in (0,1]} \mathbb{P}(|X_t| > \log(n)) \le \log^{-1}(n) \exp(-c \log^2(n))$ with $c > 0$ a constant. Then, we obtain from Equation (2.44) that

$$ 
\left\{
\begin{aligned}
\mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2\|_{n,1}^2] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m \log^2(n)}{n}} \quad [\mathbf{B}] \\
\mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma_{\mathcal{I}}^2\|_{n,1}^2] &\le \inf_{h \in S_m} \|h - \sigma_{\mathcal{I}}^2\|_n^2 + C \sqrt{\frac{m^2 \log^2(n)}{n}} \quad [\mathbf{OB}].
\end{aligned}
\right.
\quad \square $$

### 2.9.3 Proof of Section 2.4

The following lemma allows us to obtain a risk bound of $\hat{\sigma}_{m,L}^2$ defined with the empirical norm $\|\cdot\|_n$ from the risk bound defined from the pseudo norm $\|\cdot\|_{n,N}$.

**Lemma 2.9.7.** Let $\hat{\sigma}_{m,L}^2$ be the truncated projection estimator on $\mathbb{R}$ of $\sigma^2$ over the subspace $S_{m,L}$. Suppose that $L = \log^2(Nn)$, $N > 1$. Under Assumption 2.2.1, there exists a constant $C > 0$ independent of $m$ and $N$ such that

$$ \mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma^2\|_{n,N}^2] - 2\mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma^2\|_n^2] \le C \frac{m^2 \log^3(Nn)}{N}. $$

The proof of Lemma 2.9.7 is provided in [30], Theorem 3.3. The proof uses the independence of the copies $\bar{X}^1, \dots, \bar{X}^N$ of the process $X$ at discrete times, and the Bernstein inequality.

#### Proof of Theorem 2.4.1

For fixed $n$ and $N$ in $\mathbb{N}^*$, we set for all $m \in \mathcal{M}$,

$$ \Omega_{n,N,m} := \bigcap_{h \in S_m \setminus \{0\}} \left\{ \left| \frac{\|h\|_{n,N}^2}{\|h\|_n^2} - 1 \right| \le \frac{1}{2} \right\}. \quad (2.45) $$

As we can see, the empirical norms $\|h\|_{n,N}$ and $\|h\|_n$ of any function $h \in S_m \setminus \{0\}$ are equivalent on $\Omega_{n,N,m}$. More precisely, on the set $\Omega_{n,N,m}$, for all $h \in S_m \setminus \{0\}$, we have: $\frac{1}{2}\|h\|_n^2 \le \|h\|_{n,N}^2 \le \frac{3}{2}\|h\|_n^2$. We have the following result:
**Lemma 2.9.8.** Under Assumption 2.2.1, the following holds:

* If $n \ge N$ or $n \propto N$, then $m \in \mathcal{M} = \{1, \dots, \lfloor \sqrt{N}/\log(Nn) \rfloor\}$ and,
$$ \mathbb{P}(\Omega_{n,N,m}^c) \le 2 \exp(-C\sqrt{N}). $$

* If $n \le N$, then $m \in \mathcal{M} = \{1, \dots, \lfloor \sqrt{n}/\log(Nn) \rfloor\}$ and
$$ \mathbb{P}(\Omega_{n,N,m}^c) \le 2 \exp(-C\sqrt{n}). $$

where $C > 0$ is a constant.

The proof of Lemma 2.9.8 is provided in appendix.

**Proof of Theorem 2.4.1.** The proof of Theorem 2.4.1 extends the proof of Theorem 2.3.2 when $N$ tends to infinity. Then, we deduce from Equation (2.34) that

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|}^2 \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,m}} \right] \le 3 \inf_{h \in \mathcal{S}_{m,L}} \left\| h - \sigma_{|I|}^2 \right\|_n^2 + C \mathbb{E} \left( \sup_{h \in \mathcal{S}_{m,L}, \|h\|_n=1} \nu^2(h) \right) + C \Delta^2 \quad (2.46) $$

where $C > 0$ is a constant depending on $\sigma_1$, and $\nu = \nu_1 + \nu_2 + \nu_3$ with

$$ \nu_i(h) = \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} h(X_{k\Delta}^j) \zeta_{k\Delta}^{j,i}, \quad i = 1, 2, 3 $$

and the $\zeta_{k\Delta}^{j,i}$'s are the error terms depending on each path $X^j$, $j = 1, \dots, N$.

**Upper bound of** $\mathbb{E} \left( \sup_{h \in \mathcal{S}_m, \|h\|_n=1} \nu^2(h) \right)$

For all $h = \sum_{\ell=0}^{m-1} a_\ell \phi_\ell \in \mathcal{S}_m$ such that $\|h\|_n = 1$, we have $\|h\|^2 \le \frac{1}{\tau_0}$ and the coordinate vector $a = (a_0, \dots, a_{m-1})$ satisfies:

* $\|a\|_2^2 \le CK \le Cm$ for the spline basis (see [30], Lemma 2.6)

* $\|a\|_2^2 \le 1/\tau_0$ for an orthonormal basis since $\|h\|^2 = \|a\|_2^2$.

Furthermore, using the Cauchy–Schwarz inequality, we have:

$$ \nu^2(h) = \left( \sum_{\ell=0}^{m-1} a_\ell \nu(\phi_\ell) \right)^2 \le \|a\|_2^2 \sum_{\ell=0}^{m-1} \nu^2(\phi_\ell). $$

Thus, for all $\ell \in [0, m-1]$, $\nu = \nu_1 + \nu_2 + \nu_3$ and for all $i \in \{1, 2, 3\}$

$$ \mathbb{E} [\nu_i^2(\phi_\ell)] = \frac{1}{Nn^2} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \phi_\ell (X_{k\Delta}^1) \zeta_{k\Delta}^{1,i} \right)^2 \right]. $$

We finally deduce from (2.36), (2.37) and (2.38) that there exists a constant $C > 0$ depending on $\sigma_1$ such that:

$$ \mathbb{E} \left( \sup_{h \in \mathcal{S}_m, \|h\|_n=1} \nu^2(h) \right) \le C \frac{m}{Nn}. \quad (2.47) $$

We deduce from (2.46) and (2.47) that there exists a constant $C > 0$ such that,

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|}^2 \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,m}} \right] \le 3 \inf_{h \in \mathcal{S}_{m,L}} \left\| \sigma_{|I|}^2 - h \right\|_n^2 + C \left( \frac{m}{Nn} + \Delta^2 \right). \quad (2.48) $$
Since we have $\|\hat{\sigma}_m^2\|_{\infty} \le \sqrt{mL}$, then for $m$ and $L$ large enough, $\|\hat{\sigma}_m^2 - \sigma_{|I|^2}\|_{\infty}^2 \le 2mL$. There exists a constant $C > 0$ such that for all $m \in \mathcal{M}$ and for $m$ and $L$ large enough,

$$ \begin{aligned} \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \right] &= \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,m}} \right] + \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,m}^c} \right] \\ &\le \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,m}} \right] + 2mL \mathbb{P}(\Omega_{n,N,m}^c). \end{aligned} $$

Then, from Equation (2.48), Lemma 2.9.8 and for $m \in \mathcal{M} = \{1, \dots, \sqrt{\min(n, N)}/\sqrt{\log(Nn)}\}$, we have:

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \right] \le 3 \inf_{h \in S_{m,L}} \left\| h - \sigma_{|I|^2} \right\|_n^2 + C \left( \frac{m}{Nn} + mL \exp \left( -C \sqrt{\min(n, N)} \right) + \Delta^2 \right) $$

where $C > 0$ is a constant. Recall that the empirical norms $\|\cdot\|_{n,N}$ and $\|\cdot\|_n$ are equivalent on $\Omega_{n,N,m}$, that is for all $h \in S_m$, $\|h\|_n^2 \le 2\|h\|_{n,N}^2$. Thus, we have

$$ \begin{aligned} \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_n^2 \right] &= \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_n^2 \mathbb{1}_{\Omega_{n,N,m}} \right] + \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_n^2 \mathbb{1}_{\Omega_{n,N,m}^c} \right] \\ &\le \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_n^2 \mathbb{1}_{\Omega_{n,N,m}} \right] + 2mL \mathbb{P}(\Omega_{n,N,m}^c). \end{aligned} $$

For all $h \in S_{m,L} \subset S_m$, we have:

$$ \begin{aligned} \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_n^2 \mathbb{1}_{\Omega_{n,N,m}} \right] &\le 2\mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - h \right\|_n^2 \mathbb{1}_{\Omega_{n,N,m}} \right] + 2 \left\| h - \sigma_{|I|^2} \right\|_n^2 \\ &\le 4\mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - h \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,m}} \right] + 2 \left\| h - \sigma_{|I|^2} \right\|_n^2 \\ &\le 8\mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \right] + 10 \left\| h - \sigma_{|I|^2} \right\|_n^2. \end{aligned} $$

We finally conclude that

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_m^2 - \sigma_{|I|^2} \right\|_n^2 \right] \le 34 \inf_{h \in S_{m,L}} \left\| h - \sigma_{|I|^2} \right\|_n^2 + C \left( \frac{m}{Nn} + mL \exp \left( -C \sqrt{\min(n, N)} \right) + \Delta^2 \right). \quad \square $$

### Proof of Theorem 2.4.3

*Proof.* For each dimension $m \in \mathcal{M}$, we have:

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{m,L}^2 - \sigma^2 \right\|_n^2 \right] = \mathbb{E} \left[ \left\| (\hat{\sigma}_{m,L}^2 - \sigma^2) \mathbb{1}_{[-\log(N), \log(N)]} \right\|_n^2 \right] + \mathbb{E} \left[ \left\| (\hat{\sigma}_{m,L}^2 - \sigma^2) \mathbb{1}_{[-\log(N), \log(N)]^c} \right\|_n^2 \right] $$

and from Proposition 2.9.2, Lemma 2.9.4 and for $N$ large enough, there exists constants $c$, $C > 0$ such that

$$ \begin{aligned} \mathbb{E} \left[ \left\| (\hat{\sigma}_{m,L}^2 - \sigma^2) \mathbb{1}_{[-\log(N), \log(N)]^c} \right\|_n^2 \right] &\le \frac{2L}{n} \sum_{k=0}^{n-1} \mathbb{P}(|X_{k\Delta}| > \log(N)) \\ &\le 2L \sup_{t \in [0,1]} \mathbb{P}(|X_t| \ge \log(N)) \\ &\le \frac{C}{\log(N)} \exp(-c \log^2(N)). \end{aligned} $$

We deduce that

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{m,L}^2 - \sigma^2 \right\|_n^2 \right] = \mathbb{E} \left[ \left\| (\hat{\sigma}_{m,L}^2 - \sigma^2) \mathbb{1}_{[-\log(N), \log(N)]} \right\|_n^2 \right] + \frac{C}{\log(N)} \exp(-c \log^2(N)). \quad (2.49) $$

It remains to upper-bound the first term on the right hand side of Equation (2.49).
**Upper bound of** $\mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma^2\|_n^2 1_{[-\log(N),\log(N)]}]$. For all $h \in S_{m,L}$, we obtain from Equation (2.7),

$$ \gamma_{n,N}(\hat{\sigma}_{m,L}^2) - \gamma_{n,N}(\sigma^2) \leq \gamma_{n,N}(h) - \gamma_{n,N}(\sigma^2). $$

For all $h \in S_{m,L}$,

$$ \gamma_{n,N}(h) - \gamma_{n,N}(\sigma^2) = \|h - \sigma^2\|_{n,N}^2 + 2\nu_1(\sigma^2 - h) + 2\nu_2(\sigma^2 - h) + 2\nu_3(\sigma^2 - h) + 2\mu(\sigma^2 - h) $$

where

$$ \nu_i(h) = \frac{1}{nN} \sum_{j=1}^{N} \sum_{k=0}^{n-1} h(X_{k\Delta}^j) \zeta_{k\Delta}^{j,i}, \quad i \in \{1, 2, 3\}, \quad \mu(h) = \frac{1}{nN} \sum_{j=1}^{N} \sum_{k=0}^{n-1} h(X_{k\Delta}^j) R_{k\Delta}^j, $$

we deduce from Equation (A.42) that for all $h \in S_{m,L}$,

$$ \mathbb{E} [\|\hat{\sigma}_{m,L}^2 - \sigma^2\|_{n,N}^2 1_{[-\log(N),\log(N)]}] \leq \inf_{h \in S_{m,L}} \|h - \sigma^2\|_n^2 + 2 \sum_{i=1}^3 \mathbb{E}[\nu_i(\hat{\sigma}_{m,L}^2 - h)] + 2 \mathbb{E}[\mu(\hat{\sigma}_{m,L}^2 - h)]. \quad (2.50) $$

For all $i \in \{1, 2, 3\}$ and for all $h \in S_{m,L}$, one has

$$ \mathbb{E}[\nu_i(\hat{\sigma}_{m,L}^2 - h)] \leq \sqrt{2mL} \sqrt{\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_i^2(\phi_\ell)]}. \quad (2.51) $$

1. Upper bound of $\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_1^2(\phi_\ell)]$

According to Equation (A.44), we have

$$ \forall \ell \in [0, m-1], \nu_1(\phi_\ell) = \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} \phi_\ell(X_{k\Delta}^j) \zeta_{k\Delta}^{j,1} $$

where $\zeta_{k\Delta}^{j,1} = \frac{1}{\Delta} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^j) dW_s^j \right)^2 - \int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_s^j) ds \right]$ is a martingale satisfying

$$ \mathbb{E}[\zeta_{k\Delta}^{1,1} | \mathcal{F}_{k\Delta}] = 0 \quad \text{and} \quad \mathbb{E}[(\zeta_{k\Delta}^{1,1})^2 | \mathcal{F}_{k\Delta}] \leq \frac{1}{\Delta^2} \mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_s^1) ds\right)^2\right] \leq C\sigma_1^4 $$

with $C > 0$ a constant, $W = W^1$ and $(\mathcal{F}_t)_{t \ge 0}$ the natural filtration of the martingale $(M_t)_{t \in [0,1]}$ given for all $t \in [0, 1]$ by $M_t = \int_0^t \sigma(X_s^1) dW_s$. We derive that

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_1^2(\phi_\ell)] = \frac{1}{Nn^2} \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \phi_\ell(X_{k\Delta}^j) \zeta_{k\Delta}^{1,1} \right)^2 \right] = \frac{1}{Nn^2} \mathbb{E} \left[ \sum_{k=0}^{n-1} \sum_{\ell=0}^{m-1} \phi_\ell^2(X_{k\Delta}^1) (\zeta_{k\Delta}^{1,1})^2 \right]. $$

For each $k \in [0, n-1]$, we have

$$ \left\{ \begin{array}{l} \sum_{\ell=0}^{m-1} \phi_{\ell}^{2}(X_{k\Delta}^{1}) = \sum_{\ell=-M}^{K-1} B_{\ell}^{2}(X_{k\Delta}^{1}) = 1 \quad \text{for the spline basis} \\ \\ \sum_{\ell=0}^{m-1} \phi_{\ell}^{2}(X_{k\Delta}^{1}) \le Cm \quad \text{For an orthonormal basis with } C = \max_{0 \le \ell \le m-1} \|\phi_{\ell}\|_{\infty}^{2}. \end{array} \right. $$

Finally, there exists a constant $C > 0$ such that

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_1^2(\phi_\ell)] \le \begin{cases} \frac{C}{Nn} & \text{for the spline basis} \\ C \frac{m}{Nn} & \text{for an orthonormal basis.} \end{cases} $$
2. Upper bound of $\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_2^2(\phi_\ell)]$

For all $k \in [0, n-1]$ and for all $s \in [0, 1]$, set $\eta(s) = k\Delta$ if $s \in [k\Delta, (k+1)\Delta)$. We have:

$$ 
\begin{aligned}
\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_2^2(\phi_\ell)] &= \frac{4}{N} \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \int_{k\Delta}^{(k+1)\Delta} \phi_\ell(X_{k\Delta}^1) ((k+1)\Delta - s) \sigma'(X_s^1) \sigma^2(X_s^1) dW_s \right)^2 \right] \\
&= \frac{4}{N} \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \int_0^1 \phi_\ell(X_{\eta(s)}^1) (\eta(s) + \Delta - s) \sigma'(X_s^1) \sigma^2(X_s^1) dW_s \right)^2 \right].
\end{aligned}
$$

We conclude that

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_2^2(\phi_\ell)] \le \begin{cases} \frac{C}{Nn^2} & \text{for the spline basis} \\ C \frac{m}{Nn^2} & \text{for an orthonormal basis.} \end{cases} $$

where the constant $C > 0$ depends on the diffusion coefficient and the upper bound of the basis functions.

3. Upper bound of $\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_2^3(\phi_\ell)]$

We have:

$$ 
\begin{aligned}
\sum_{\ell=0}^{m-1} \mathbb{E}[\nu_3^2(\phi_\ell)] &= \frac{4}{Nn^2} \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \int_{k\Delta}^{(k+1)\Delta} \phi_\ell(X_{k\Delta}^1) b(X_{k\Delta}^1) \sigma(X_s^1) dW_s \right)^2 \right] \\
&= \frac{4}{Nn^2} \sum_{\ell=0}^{m-1} \mathbb{E} \left[ \left( \int_0^1 \phi_\ell(X_{\eta(s)}^1) b(X_{\eta(s)}^1) \sigma(X_s^1) dW_s \right)^2 \right] \\
&\le \frac{4}{Nn^2} \mathbb{E} \left[ \int_0^1 \sum_{\ell=0}^{m-1} \phi_\ell^2(X_{\eta(s)}^1) b(X_{\eta(s)}^1) \sigma^2(X_s^1) ds \right].
\end{aligned}
$$

Since for all $x \in \mathbb{R}$, $b(x) \le C_0(1+x^2)$ and $\sup_{t \in [0,1]} \mathbb{E}(|X_t|^2) < \infty$, there exists a constant $C > 0$ depending on the diffusion coefficient such that

$$ \sum_{\ell=0}^{m-1} \mathbb{E}[\nu_3^2(\phi_\ell)] \le \begin{cases} \frac{C}{Nn^2} & \text{for the spline basis} \\ C \frac{m}{Nn^2} & \text{for an orthonormal basis.} \end{cases} $$

We finally deduce that from Equations (2.50) and (2.51) that for all $h \in S_{m,L}$,

$$ 
\left\{ 
\begin{aligned}
\mathbb{E} \left[ \left\| \hat{\sigma}_{m,L}^2 - \sigma^2 \right\|_{n,N}^2 \mathbf{1}_{[-\log(N), \log(N)])} \right] &\le \inf_{h \in S_{m,L}} \|h - \sigma^2\|_n^2 + C \sqrt{\frac{mL}{Nn}} + 2\mathbb{E} \left[ \mu(\hat{\sigma}_{m,L}^2 - h) \right] \quad (1) \\
\mathbb{E} \left[ \left\| \hat{\sigma}_{m,L}^2 - \sigma^2 \right\|_{n,N}^2 \mathbf{1}_{[-\log(N), \log(N))]} \right] &\le \inf_{h \in S_{m,L}} \|h - \sigma^2\|_n^2 + C \sqrt{\frac{m^2L}{Nn}} + 2\mathbb{E} \left[ \mu(\hat{\sigma}_{m,L}^2 - h) \right] \quad (2)
\end{aligned}
\right. 
\quad (2.52) $$

where $C > 0$ is a constant, the result (1) corresponds to the spline basis, and the result (2) corresponds to any orthonormal basis. It remains to obtain an upper bound of the term $\mu(\hat{\sigma}_{m,L}^2 - h)$. For all $a > 0$ and for all $h \in S_{m,L}$,

$$ 2\mu(\hat{\sigma}_{m,L}^2 - h) \le \frac{2}{a} \|\hat{\sigma}_{m,L}^2 - \sigma^2\|_{n,N}^2 + \frac{2}{a} \|h - \sigma^2\|_{n,N}^2 + \frac{a}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} (R_{k\Delta}^j)^2 $$

$$ 2\mathbb{E}[\mu(\hat{\sigma}_{m,L}^2 - h)] \le \frac{2}{a} \mathbb{E} \|\hat{\sigma}_{m,L}^2 - \sigma^2\|_{n,N}^2 + \frac{2}{a} \inf_{h \in S_{m,L}} \|h - \sigma^2\|_n^2 + \frac{a}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} \mathbb{E} \left[ (R_{k\Delta}^j)^2 \right]. $$
Using Equations (2.33), (2.52) and setting $a = 4$, we deduce that there exists a constant $C > 0$ depending on $\sigma_1$ such that,

$$ \left\{ \begin{array}{l} \mathbb{E} \left[ \left\| \hat{\sigma}_{m,L}^2 - \sigma^2 \right\|_{n,N}^2 \mathbf{1}_{[-\log(N), \log(N)]]} \right] \le \inf_{h \in S_{m,L}} \|h - \sigma^2\|_n^2 + C \left( \sqrt{\frac{mL}{Nn}} + \Delta^2 \right) [\mathbf{B}] \\ \\ \mathbb{E} \left[ \left\| \hat{\sigma}_{m,L}^2 - \sigma^2 \right\|_{n,N}^2 \mathbf{1}_{[-\log(N), \log(N)]]} \right] \le \inf_{h \in S_{m,L}} \|h - \sigma^2\|_n^2 + C \left( \sqrt{\frac{m^2L}{Nn}} + \Delta^2 \right) [\mathbf{H}]. \end{array} \right. \quad (2.53) $$

The final result is obtained from Equations (2.49) and (2.53). □

### 2.9.4 Proof of Section 2.5

#### 2.9.5 Proof of Lemma 2.5.1

*Proof.* It is proven in [18] that for each dimension $m \in \mathcal{M}$, the Gram matrix $\Psi_m$ built from the Hermite basis is invertible. For the case of the **B-spline** basis, let us consider a vector $(x_{-M}, \cdots, x_{K-1}) \in \mathbb{R}^m$ such that $x_j \in [u_{j+M}, u_{j+M+1})$ and $B_j(x_j) \ne 0$. Since $[u_{j+M}, u_{j+M+1}) \cap [u_{j'+M}, u_{j'+M+1}) = \emptyset$ for all $j, j' \in \{-M, \cdots, K-1\}$ such that $j \ne j'$, then for all $j, j' \in \{-M, \cdots, K-1\}$ such that $j \ne j'$, $B_j(x_{j'}) = 0$. Consequently, we obtain:

$$ \begin{aligned} \det \left( (B_{\ell}(x_{\ell'}))_{-M \le \ell, \ell' \le K-1} \right) &= \det (\operatorname{diag}(B_{-M}(x_M), \cdots, B_{K-1}(x_{K-1}))) \\ &= \prod_{\ell=-M}^{K-1} B_{\ell}(x_{\ell}) \ne 0. \end{aligned} $$

Then, we deduce from [18], Lemma 1 that the matrix $\Psi_m$ is invertible for all $m \in \mathcal{M}$, where the function $f_T$ are replaced by

$$ f_n : x \mapsto \frac{1}{n} \sum_{k=0}^{n-1} p_X(k\Delta, x) $$

with $\lambda([-A_N, A_N] \cap \operatorname{supp}(f_n)) > 0$, $\lambda$ being the Lebesgue measure.

**Case of the B-spline basis.** For all $w \in \mathbb{R}^m$ such that $\|w\|_{2,m} = 1$, we have:

$$ w' \Psi_m w = \|t_w\|_n^2 = \int_{-A_N}^{A_N} t_w^2(x) f_n(x) dx + \frac{t_w^2(x_0)}{n} \quad \text{with} \quad t_w = \sum_{\ell=-M}^{K-1} w_\ell B_\ell. $$

Under Assumption 2.2.1, the transition density $(t, x) \mapsto p_X(t, x)$ is approximated as follows

$$ \forall (t, x) \in (0, 1] \times \mathbb{R}, \quad \frac{1}{K_* \sqrt{t}} \exp \left( -c_\sigma \frac{x^2}{t} \right) \le p_X(t, x) \le \frac{K_*}{\sqrt{t}} \exp \left( -\frac{x^2}{c_\sigma t} \right) $$

where $K_* > 1$ and $c_\sigma > 1$. Since $s \mapsto \exp(-c_\sigma x^2/s)$ is an increasing function, then for $n$ large enough and for all $x \in [-A_N, A_N]$,

$$ \begin{aligned} f_n(x) &\ge \frac{1}{K_* n} \sum_{k=1}^{n-1} \exp \left( -c \frac{x^2}{k \Delta} \right) \ge \frac{1}{K_*} \int_0^{1-\Delta} \exp \left( -c_\sigma \frac{x^2}{s} \right) ds \\ &\ge \frac{1}{K_*} \int_{1-(\log(N))^{-1}}^{1-(2\log(N))^{-1}} \exp \left( -c_\sigma \frac{x^2}{s} \right) ds \\ &\ge \frac{1}{2K_* \log(N)} \exp \left( -\frac{c_\sigma x^2}{1 - \log^{-1}(N)} \right). \end{aligned} $$
Thus, the density function satisfies

$$ \forall x \in [-A_N, A_N], \quad f_n(x) \ge \frac{1}{2K_* \log(N)} \exp\left(-\frac{c_\sigma A_N^2}{1 - \log^{-1}(N)}\right) \ge \frac{1}{2K_* \log(N)} \exp(-c_\sigma A_N^2). $$

Finally, since there exists a constant $C_1 > 0$ such that $\|t_w\|^2 \ge C_1 A_N K_N^{-1}$ (see [30], Lemma 2.6), for all $w \in \mathbb{R}^m$ ($m = K_N + M$) such that $\|w\|_{2,m} = 1$, there exists a constant $C > 0$ such that,

$$ w'\Psi_m w \ge \frac{CA_N}{m \log(N)} \exp(-c_\sigma A_N^2). $$

**Case of the Hermite basis.** For all $w \in \mathbb{R}^m$ such that $\|w\|_{2,m} = 1$, we have

$$ w'\Psi_m w = \|t_w\|_n^2 = \int_{-\infty}^{+\infty} t_w^2(x) f_n(x) dx + \frac{t_w^2(x_0)}{n} \quad \text{with} \quad t_w = \sum_{\ell=0}^{m-1} w_\ell h_\ell. $$

Recall that for all $x \in \mathbb{R}$ such that $|x| \ge \sqrt{(3/2)(4m+3)}$, $|h_\ell(x)| \le c|x| \exp(-c_0 x^2)$ for all $\ell \ge 0$. Then we have

$$ \begin{aligned} w'\Psi_m w &\ge \int_{|x|\le \sqrt{(3/2)(4m+3)}} \left(\sum_{\ell=0}^{m-1} w_\ell h_\ell(x)\right)^2 f_n(x) dx \\ &\ge \inf_{x\in[-\sqrt{(3/2)(4m+3)}, \sqrt{(3/2)(4m+3)}]} f_n(x) \int_{|x|\le \sqrt{(3/2)(4m+3)}} \left(\sum_{\ell=0}^{m-1} w_\ell h_\ell(x)\right)^2 dx \\ &\ge \frac{1}{2K_* \log(N)} \exp\left(-\frac{3c_\sigma(4m+3)}{2(1-\log^{-1}(N))}\right) \int_{|x|\le \sqrt{(3/2)(4m+3)}} \left(\sum_{\ell=0}^{m-1} w_\ell h_\ell(x)\right)^2 dx \end{aligned} $$

since for all $x \in \mathbb{R}$, $f_n(x) \ge (1/2K_* \log(N)) \exp(-\frac{c_\sigma x^2}{1-\log^{-1}(N)})$. Set $a_N = \sqrt{(3/2)(4m+3)}$, then we obtain

$$ \begin{aligned} w'\Psi_m w &\ge \frac{\exp\left(-\frac{3c_\sigma(4m+3)}{2(1-\log^{-1}(N))}\right)}{2K_* \log(N)} \left(\int_{-\infty}^{+\infty} \left(\sum_{\ell=0}^{m-1} w_\ell h_\ell(x)\right)^2 dx - \int_{|x|>a_N} \left(\sum_{\ell=0}^{m-1} w_\ell h_\ell(x)\right)^2 dx\right) \\ &\ge \frac{\exp\left(-\frac{3c_\sigma(4m+3)}{2(1-\log^{-1}(N))}\right)}{2K_* \log(N)} \left(1 - 2c^2m \int_{a_N}^{+\infty} x^2 \exp(-8c_0 x^2) dx\right) \\ &\ge \frac{\exp\left(-\frac{3c_\sigma(4m+3)}{2(1-\log^{-1}(N))}\right)}{2K_* \log(N)} \left(1 - \frac{c^2m}{8c_0} \sqrt{\frac{3}{2}(4m+3)} \exp(-12c_0(4m+3))\right) \end{aligned} $$

where $c, c_0 > 0$ are constants depending on the Hermite basis. Finally, for $N$ large enough,

$$ 1 - \frac{c^2 m}{8c_0} \sqrt{\frac{3}{2}(4m+3)} \exp(-12c_0(4m+3)) \ge \frac{1}{2}. $$

Finally, there exists a constant $C > 0$ such that for all $w \in \mathbb{R}^m$ such that $\|w\|_{2,m}$,

$$ w'\Psi_m w \ge \frac{C}{\log(N)} \exp\left(-\frac{3c_\sigma(4m+3)}{2(1-\log^{-1}(N))}\right). $$

□
2.9.6 Proof of Theorem 2.5.2

The proof of Theorem 2.5.2 relies on the following lemma:

**Lemma 2.9.9.** Under Assumptions 2.2.1 and for $\sigma^2 \in \Sigma_I(\beta, R)$ with $\beta \ge 1$, $I = [-A_N, A_N]$ and

$$N \propto n, A_N = o\left(\sqrt{\log(N)}\right), K \propto \left((Nn)^{1/(2\beta+1)}A_N\right) \quad (m = K+M),$$

the following holds:

$$\mathbb{P}(\Omega_{n,N,m}^c) \le C \exp(-c \log^{3/2}(N))$$

where c, C > 0 are constants independent of N.

*Proof of Theorem 2.5.2.* According to Equation (55) in the proof of Theorem 4.1, for all dimension $m = K + M$, with $K \in \mathcal{M}$, and for all $h \in S_{K+M}$, there exists a constant $C > 0$ such that

$$\mathbb{E} \left[ \left\| \hat{\sigma}_{A_N, m}^2 - \sigma_{A_N}^2 \right\|_{n,N}^2 \mathbf{1}_{\Omega_{n,N,m}} \right] \le C \left[ \inf_{h \in S_{K+M,L}} \|h - \sigma_{A_N}^2\|_n^2 + \mathbb{E} \left( \sup_{h \in S_{K+M}, \|h\|_n = 1} \nu^2(h) \right) + \Delta^2 \right] \quad (2.54)$$

where $\Omega_{n,N,m}$ is given in Equation (52) and $\nu = \nu_1 + \nu_2 + \nu_3$ with the $\nu_i$ given in Equation (32).

For all $h = \sum_{\ell=-M}^{K-1} a_\ell B_\ell \in S_{K+M,L_N'}$

$$\|h\|_n^2 = \mathbb{E} \left[ \frac{1}{n} \sum_{k=0}^{n-1} h^2(X_{k\Delta}) \right] = \sum_{\ell=-M}^{K-1} \sum_{\ell'=-M}^{K-1} a_\ell a_{\ell'} \mathbb{E} \left[ \frac{1}{n} \sum_{k=0}^{n-1} B_\ell(X_{k\Delta}) B_{\ell'}(X_{k\Delta}) \right] = a' \Psi_m a.$$

The Gram matrix $\Psi_m$ is invertible for each $K \in \mathcal{M}$ (see proof of Lemma 2.5.1). It follows that for all $h = \sum_{\ell=-M}^{K-1} a_\ell B_\ell$ such that $\|h\|_n^2 = a'\Psi_m a = 1$, one has $a = \Psi_m^{-1/2}u$ where $u \in \mathbb{R}^m$ and $\|u\|_{2,m} = 1$. Furthermore, we have:

$$h = \sum_{\ell=-M}^{K-1} a_\ell B_\ell = \sum_{\ell=-M}^{K-1} u_\ell \sum_{\ell'} \left[ \Psi_m^{-1/2} \right]_{\ell',\ell} B_{\ell'}.$$

Then for all $h \in S_{K+M}$, we have $\nu^2(h) \le 3(\nu_1^2(h) + \nu_2^2(h) + \nu_3^2(h))$ where,

$$\forall i \in \{1, 2, 3\}, \nu_i^2(h) \le \sum_{\ell=-M}^{K-1} \left( \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} \sum_{\ell'==-M}^{K-1} \left[ \Psi_m^{-1/2} \right]_{\ell',\ell} B_{\ell'}(X_{k\Delta}^j) \zeta_{k\Delta}^{j,i} \right)^2.$$

So we obtain,

$$\forall i \in \{1, 2, 3\}, \quad \mathbb{E} \left[ \sup_{h \in S_{K+M}, \|h\|_n=1} \nu_i^2(h) \right] \le \frac{1}{Nn^2} \sum_{\ell=-M}^{K-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \sum_{\ell'=-M}^{K-1} \left[ \Psi_m^{-1/2} \right]_{\ell',\ell} B_{\ell'}(X_{k\Delta}^j) \zeta_{k\Delta}^{1,i} \right)^2 \right].$$

For $i = 1$, we have $\zeta_{k\Delta}^{1,1} = \frac{1}{\Delta} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s \right)^2 - \int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_s^1) ds \right]$ and we obtained in the proof of Theorem 3.2 that there exists a constant $C > 0$ such that for all $k \in [0, n-1]$,

$$\mathbb{E}[\zeta_{k\Delta}^{1,1}|\mathcal{F}_{k\Delta}] = 0, \quad \mathbb{E}[(\zeta_{k\Delta}^{1,1})^2 | \mathcal{F}_{k\Delta}] \le C \mathbb{E} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_u) du \right)^2 \right] \le C \sigma_1^4 \Delta^2.$$

We deduce that

$$\begin{align*}
\mathbb{E} \left[ \sup_{h \in S_{K+M}, \|h\|_n=1} \nu_1^2(h) \right] &= \frac{1}{Nn^2 \Delta^2} \sum_{\ell=0}^{K-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \sum_{\ell'=-M}^{K-1} \left[ \Psi_m^{-1/2} \right]_{\ell',\ell} B_{\ell'}(X_{k\Delta}^1) \zeta_{k\Delta}^{1,1} \right)^2 \right] \\
&\le \frac{1}{N} \sum_{\ell=-M}^{K-1} \sum_{k=0}^{n-1} \mathbb{E} \left\{ \left( \sum_{\ell'=-M}^{K-1} \left[ \Psi_m^{-1/2} \right]_{\ell',\ell} B_{\ell'}(X_{k\Delta}^1) \right)^2 \left( \zeta_{k\Delta}^{1,1} \right)^2 \right\} \\
&\le \frac{4\sigma_1^2}{Nn} \sum_{\ell=-M}^{K-1} \sum_{\ell'=-M}^{K-1} \sum_{\ell''=-M}^{K-1} \left[ \Psi_m^{-1/2} \right]_{\ell',\ell} \left[ \Psi_m^{-1/2} \right]_{\ell'',\ell} \left[ \Psi_m^{-1/2} \right]_{\ell',\ell''}.
\end{align*}$$
We have:

$$ \sum_{\ell=-M}^{K-1} \sum_{\ell'=-M}^{K-1} \sum_{\ell''=-M}^{K-1} [\Psi_m^{-1/2}]_{\ell',\ell} [\Psi_m^{-1/2}]_{\ell'',\ell} [\Psi_m^{-1/2}]_{\ell',\ell''} = \mathrm{Tr}(\Psi_m^{-1}\Psi_m) = \mathrm{Tr}(I_m) = m. $$

So we obtain

$$ \mathbb{E} \left[ \sup_{h \in S_{K+M}} \nu_1^2(h) \right] \le \frac{4\sigma_1^2 m}{Nn}. $$

For $i=2$, we have $\zeta_{k\Delta}^{1,2} = \frac{2}{\Delta} \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s) \sigma'(X_s^1)\sigma^2(X_s^1) dW_s$ and

$$ \begin{aligned} \mathbb{E} \left[ \sup_{h \in S_{K+M}, \|h\|_n = 1} \nu_2^2(h) \right] &\le \frac{1}{Nn^2} \sum_{\ell=-M}^{K-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \sum_{\ell'=-M}^{K-1} [\Psi_m^{-1/2}]_{\ell',\ell} B_{\ell'}(X_{k\Delta}^j) \zeta_{k\Delta}^{1,2} \right)^2 \right] \\ &\le \frac{4\sigma_1^4 \|\sigma'\|_{\infty}^2 \Delta}{Nn^2} \sum_{\ell=-M}^{K-1} \sum_{k=0}^{n-1} \mathbb{E} \left[ \left( \sum_{\ell'=-M}^{K-1} [\Psi_m^{-1/2}]_{\ell',\ell} B_{\ell'}(X_{k\Delta}^1) \right)^2 \right] \\ &\le \frac{4\sigma_1^2 \|\sigma'\|_{\infty}^2 m}{Nn^2}. \end{aligned} $$

For $i=3$, we have $\zeta_{k\Delta}^{1,3} = 2b(X_{k\Delta}^1) \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^1) dW_s$ and there exists constants $C_1, C_2 > 0$ such that

$$ \begin{aligned} \mathbb{E} \left[ \sup_{h \in S_{K+M}, \|h\|_n = 1} \nu_3^2(h) \right] &\le \frac{1}{Nn^2} \sum_{\ell=-M}^{K-1} \mathbb{E} \left[ \left( \sum_{k=0}^{n-1} \sum_{\ell'=-M}^{K-1} [\Psi_m^{-1/2}]_{\ell',\ell} B_{\ell'}(X_{k\Delta}^j) \zeta_{k\Delta}^{1,3} \right)^2 \right] \\ &\le C_1 \frac{\sigma_1^2 \Delta}{Nn^2} \sum_{\ell=-M}^{K-1} \sum_{k=0}^{n-1} \mathbb{E} \left[ \left( \sum_{\ell'=-M}^{K-1} [\Psi_m^{-1/2}]_{\ell',\ell} B_{\ell'}(X_{k\Delta}^1) \right)^2 \right] \\ &\le C_2 \frac{\sigma_1^2 m}{Nn^2}. \end{aligned} $$

Finally, there exists a constant $C > 0$ depending on $\sigma_1$ and $M$ such that

$$ \mathbb{E} \left[ \sup_{h \in S_{K+M}, \|h\|_n = 1} \nu^2(h) \right] \le C \frac{m}{Nn}. \qquad (2.55) $$

From Equations (2.54) and (2.55), we deduce that

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{A,N,m}^2 - \sigma_{A,N}^2 \right\|_{n,N}^2 \mathbf{1}_{\Omega_{n,N,m}} \right] \le C \left( \inf_{h \in S_{K+M,L}} \|h - \sigma_{A,N}^2\|_n^2 + \frac{m}{Nn} + \Delta^2 \right) $$

where $C > 0$ is a constant depending on $\sigma_1$ and $M$. We obtain

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{A,N,m}^2 - \sigma_{A,N}^2 \right\|_{n,N}^2 \right] \le C \left( \inf_{h \in S_{K+M,L}} \|h - \sigma_{A,N}^2\|_n^2 + \frac{m}{Nn} + \Delta^2 \right) + \mathbb{E} \left[ \left\| \hat{\sigma}_{A,N,m}^2 - \sigma_{A,N}^2 \right\|_{n,N}^2 \mathbf{1}_{\Omega_{n,N,m}^c} \right] $$

and for $N$ large enough, $\left\| \hat{\sigma}_{A,N,m}^2 - \sigma_{A,N}^2 \right\|_{n,N}^2 \le 4mL$, and according to Lemma 2.9.9,

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{A,N,m}^2 - \sigma_{A,N}^2 \right\|_{n,N}^2 \mathbf{1}_{\Omega_{n,N,m}^c} \right] \le 4mL \mathbb{P} \left( \Omega_{n,N,m}^c \right) \le CmL \exp \left( -c \log^{3/2}(N) \right) $$

where $c > 0$ is a constant. Thus, there exists a constant $C > 0$ such that

$$ \begin{aligned} \mathbb{E} \left[ \left\| \hat{\sigma}_{A,N,m}^2 - \sigma_{A,N}^2 \right\|_{n,N}^2 \right] &\le \mathbb{E} \left[ \left\| \hat{\sigma}_{A,N,m}^2 - \sigma_{A,N}^2 \right\|_{n,N}^2 \mathbf{1}_{\Omega_{n,N,m}} \right] + \mathbb{E} \left[ \left\| \hat{\sigma}_{A,N,m}^2 - \sigma_{A,N}^2 \right\|_{n,N}^2 \mathbf{1}_{\Omega_{n,N,m}^c} \right] \\ &\le C \left( \inf_{h \in S_{K+M,L}} \|h - \sigma_{A,N}^2\|_n^2 + \frac{m}{Nn} + mL \exp(-c \log^{3/2}(N)) + \Delta^2 \right). \end{aligned} $$
Then, as $n \propto N$ and $L = \log(N)$, there exists a constant $C > 0$ such that

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{A_N, m}^2 - \sigma_{A_N}^2 \right\|_{n,N}^2 \right] \le C \left( \inf_{h \in S_{K+M,L}} \left\| h - \sigma_{A_N}^2 \right\|_n^2 + \frac{m}{Nn} \right). $$

Finally, since $\sigma^2 \in \Sigma_I(\beta, R)$ with $\beta \ge 1$ and $I = [-A_N, A_N]$, one has

$$ \inf_{h \in S_{K+M,L}} \|h - \sigma_{A_N}^2\|_n^2 \le C A_N^{2\beta} K^{-2\beta} $$

where the constant $C > 0$ depends on $\beta, R$ and $M$. Furthermore, as we chose the interval $[-A_N, A_N]$ such that $A_N = o(\sqrt{\log(N)})$ and for $K \propto ((Nn)^{1/(2\beta+1)}A_N)$, we obtain

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{A_N, m}^2 - \sigma_{A_N}^2 \right\|_{n,N}^2 \right] \le C \log^\beta(N) (Nn)^{-2\beta/(2\beta+1)}. $$

□

### 2.9.7 Proof of Section 2.6

#### Proof of Theorem 2.6.1

Set for all $K, K' \in \mathcal{K} = \{2^q, q = 0, \dots, q_{\max}, 2^{q_{\max}} \le \sqrt{N}/\log(N)\} \subset \mathcal{M}$,

$$ \mathcal{T}_{K,K'} = \{g \in \mathcal{S}_{K+M} + \mathcal{S}_{K'+M}, \|g\|_n = 1, \|g\|_{\infty} \le \sqrt{L}\}. \quad (2.56) $$

Recall that for all $j \in [1, N]$ and for all $k \in [0, n]$,

$$ \zeta_{k\Delta}^{j,1} = \frac{1}{\Delta} \left[ \left( \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s^j) dW_s^j \right)^2 - \int_{k\Delta}^{(k+1)\Delta} \sigma^2(X_s^j) ds \right]. $$

The proof of Theorem 2.6.1 relies on the following lemma whose proof is in Appendix.

**Lemma 2.9.10.** Under Assumption 2.2.1, for all $\varepsilon, v > 0$ and $g \in \mathcal{T}_{K,K'}$, there exists a real constant $C > 0$ such that,

$$ \mathbb{P} \left( \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} g(X_{k\Delta}^j) \zeta_{k\Delta}^{j,1} \ge \varepsilon, \|g\|_{n,N}^2 \le v^2 \right) \le \exp \left( -C \frac{Nn\varepsilon^2}{\sigma_1^2 (\varepsilon \|g\|_{\infty} + 4\sigma_1^2 v^2)} \right) $$

and for all $x > 0$ such that $x \le \varepsilon^2/\sigma_1^2 (\varepsilon \|g\|_{\infty} + 4\sigma_1^2 v^2)$,

$$ \mathbb{P} \left( \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} g(X_{k\Delta}^j) \zeta_{k\Delta}^{j,1} \ge 2\sigma_1^2 v \sqrt{x} + \sigma_1^2 \|g\|_{\infty} x, \|g\|_{n,N}^2 \le v^2 \right) \le \exp(-CNnx). $$

**Proof of Theorem 2.6.1.** From Equation (2.17), we have

$$ \widehat{K} := \arg \min_{K \in \mathcal{K}} \{ \gamma_{n,N}(\widehat{\sigma}_K^2) + \text{pen}_1(K) \}. $$

For all $K \in \mathcal{K}$ and $h \in S_{K+M,L}$,

$$ \gamma_{n,N}(\widehat{\sigma}_K^2) + \text{pen}_1(\widehat{K}) \le \gamma_{n,N}(h) + \text{pen}_1(K), $$

then, for all $K \in \mathcal{K}$ and for all $h \in S_{K+M,L}$,

$$ \begin{align*} \gamma_{n,N}(\widehat{\sigma}_K^2) - \gamma_{n,N}(\sigma_I^2) &\le \gamma_{n,N}(h) - \gamma_{n,N}(\sigma_I^2) + \text{pen}_1(K) - \text{pen}_1(\widehat{K}) \\ \left\| \widehat{\sigma}_K^2 - \sigma_I^2 \right\|_{n,N}^2 &\le \left\| h - \sigma_I^2 \right\|_{n,N}^2 + 2\nu (\widehat{\sigma}_K^2 - h) + 2\mu (\widehat{\sigma}_K^2 - h) + \text{pen}_1(K) - \text{pen}_1(\widehat{K}) \\ &\le \left\| h - \sigma_I^2 \right\|_{n,N}^2 + \frac{1}{d} \left\| \widehat{\sigma}_K^2 - t \right\|_n^2 + d \sup_{g \in \mathcal{T}_{K,\widehat{K}}} \nu^2(g) + \frac{1}{d} \left\| \widehat{\sigma}_K^2 - h \right\|_{n,N}^2 \\ &\quad + \frac{d}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} (R_{k\Delta}^j)^2 + \text{pen}_1(K) - \text{pen}_1(\widehat{K}) \end{align*} $$
where $d > 1$ and the space $\mathcal{T}_{K,\hat{K}}$ is given in Equation (2.56). On the set $\Omega_{n,N,K_{\max}}$ (given in Equation (2.45)): $\forall h \in S_{K+M}$, $\frac{1}{2}\|h\|_n^2 \le \|h\|_{n,N}^2 \le \frac{3}{2}\|h\|_n^2$. Then on $\Omega_{n,N,K_{\max}}$, for all $d > 1$ and for all $h \in S_{K+M}$ with $K \in \mathcal{K}$,

$$ \left(1 - \frac{10}{d}\right) \left\|\hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2}\right\|_{n,N}^2 \le \left(1 + \frac{10}{d}\right) \left\|h - \sigma_{|I|^2}\right\|_{n,N}^2 + d \sup_{h \in \mathcal{T}_{K,\hat{K}}} \nu^2(h) + \frac{d}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} (R_{k\Delta}^j)^2 \\ + \text{pen}_1(K) - \text{pen}_1(\hat{K}). $$

We set $d = 20$. Then, on $\Omega_{n,N,\max}$ and for all $h \in S_{K+M,L}$,

$$ \left\|\hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2}\right\|_{n,N}^2 \le 3 \left\|h - \sigma_{|I|^2}\right\|_{n,N}^2 + 20 \sup_{h \in \mathcal{T}_{K,\hat{K}}} \nu^2(h) + \frac{20}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} (R_{k\Delta}^j)^2 + 2 (\text{pen}_1(K) - \text{pen}_1(\hat{K})). \quad (2.57) $$

Let $q_1: \mathcal{K}^2 \to \mathbb{R}_+$ such that $160q_1(K, K') \le 18\text{pen}_1(K) + 16\text{pen}_1(K')$. Thus, on the set $\Omega_{n,N,K_{\max}}$, there exists a constant $C > 0$ such that for all $h \in S_{K+M}$

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,K_{\max}}} \right] \le 34 \left( \inf_{h \in S_{K+M,L}} \left\| h - \sigma_{|I|^2} \right\|_n^2 + \text{pen}_1(K) \right) \\ + 160 \mathbb{E} \left( \sup_{h \in \mathcal{T}_{K,\hat{K}}} \nu_1^2(h) - q_1(K, \hat{K}) \right) + C \Delta^2 $$

where

$$ \nu_1(h) := \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} h(X_{k\Delta}^j) \zeta_{k\Delta}^{j,1} \quad (2.58) $$

with $\zeta_{k\Delta}^{j,1}$ the error term. We set for all $K, K' \in \mathcal{K}$,

$$ G_K(K') := \sup_{h \in \mathcal{T}_{K,K'}} \nu_1^2(h) $$

and for $N$ and $n$ large enough, $\left\|\hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2}\right\|_{n,N}^2 \le 4(K + M)L$. We deduce that,

$$ \begin{aligned} \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \right] &\le \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,K_{\max}}} \right] + \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2} \right\|_{n,N}^2 \mathbb{1}_{\Omega_{n,N,K_{\max}}^c} \right] \\ &\le 34 \inf_{K \in \mathcal{K}} \left( \inf_{h \in S_{K+M}} \left\| h - \sigma_{|I|^2} \right\|_n^2 + \text{pen}_1(K) \right) \\ &\quad + C \Delta^2 + 4(K+M)L \mathbb{P}(\Omega_{n,N,K_{\max}}^c) \\ &\quad + 160 \mathbb{E} \left[ \left( G_K(\hat{K}) - q_1(K, \hat{K}) \right)_+ \mathbb{1}_{\Omega_{n,N,K_{\max}}} \right]. \end{aligned} $$

The unit ball $\bar{B}_{\|\cdot\|_n}(0,1)$ of the approximation subspace $S_{K+M}$ with respect to norm $\|\cdot\|_n$ defined as follows:

$$ \bar{B}_{\|\cdot\|_n}(0,1) = \{h \in S_{K+M} : \|h\|_n \le 1\} = \left\{h \in S_{K+M} : \|h\| \le \frac{1}{\tau_0}\right\} = \bar{B}_2(0, 1/\tau_0), $$

admits an $\varepsilon$-net $E_\varepsilon$ such that for each $\varepsilon \in (0,1]$, $|E_\varepsilon| \le \left(\frac{3}{\varepsilon\tau_0}\right)^{K+M}$ (see [74], Proposition 1.3, page 487).

Recall that $\mathcal{T}_{K,K'} = \{g \in S_{K+M} + S_{K'+M}, \|g\|_n = 1, \|g\|_{\infty} \le \sqrt{L}\}$ and consider the sequence $(E_{\varepsilon_k})_{k \ge 1}$ of $\varepsilon$-net with $\varepsilon_k = \varepsilon_0 2^{-k}$ and $\varepsilon_0 \in (0,1]$. Moreover, set $N_k = \log(|E_{\varepsilon_k}|)$ for each $k \ge 0$. Then for each $g \in S_{K+M} + S_{K'+M}$ such that $\|g\|_{\infty} \le \sqrt{L}$, there exists a sequence $(g_k)_{k \ge 0}$ with $g_k \in E_{\varepsilon_k}$ such that $g = g_0 + \sum_{k=1}^{\infty} g_k - g_{k-1}$. Set $\tilde{\mathbb{P}} := \mathbb{P}(\cdot \cap \Omega_{n,N,K_{\max}})$ and

$$ \tau := \sigma_1^2 \sqrt{6x_0^{n,N}} + \sigma_1^2 \sqrt{L} x_0^{n,N} + \sum_{k \ge 1} \varepsilon_{k-1} \left\{ \sigma_1^2 \sqrt{6x_k^{n,N}} + 2\sigma_1^2 \sqrt{L} x_k^{n,N} \right\} = y_0^{n,N} + \sum_{k \ge 0} y_k^{n,N}. $$
For all $h \in \mathcal{T}_{K,K'}$ and on the event $\Omega_{n,N,K_{\max}}$, one has $\|h\|_{n,N}^2 \le \frac{3}{2}\|h\|_n^2 = \frac{3}{2}$. Then, using the chaining technique of [7], we have

$$
\begin{aligned}
\tilde{\mathbb{P}} \left( \sup_{h \in \mathcal{T}_{K,K'}} \nu_1(h) > \tau \right) &= \tilde{\mathbb{P}} \left( \exists (h_k)_{k \ge 0} \in \prod_{k \ge 0} E_{\varepsilon_k} / \nu_1(h) = \nu_1(h_0) + \sum_{k=1}^{\infty} \nu_1(h_k - h_{k-1}) > \tau \right) \\
&\le \sum_{h_0 \in E_0} \tilde{\mathbb{P}} (\nu_1(h_0) > y_0^{n,N}) + \sum_{k=1}^{\infty} \sum_{\substack{h_k \in E_{\varepsilon_k} \\ h_{k-1} \in E_{\varepsilon_{k-1}}}} \tilde{\mathbb{P}} (\nu_1(h_k - h_{k-1}) > y_k^{n,N}).
\end{aligned}
$$

According to Equation (2.58) and Lemma 2.9.10, there exists a constant $C > 0$ such that

$$
\begin{aligned}
\tilde{\mathbb{P}}(\nu_1(h_0) > y_0^{n,N}) &\le \tilde{\mathbb{P}}(\nu_1(h_0) > \sigma_1 \sqrt{6x_0^{n,N}} + \sigma_1^2 \|h_0\|_{\infty} x_0^{n,N}) \\
&\le \exp(-CNnx_0^{n,N}), \\
\forall k \ge 1, \tilde{\mathbb{P}}(\nu_1(h_k - h_{k-1}) > y_0^{n,N}) &\le \tilde{\mathbb{P}}(\nu_1(h_k - h_{k-1}) > \sigma_1 \sqrt{6x_k^{n,N}} + \sigma_1^2 \|h_k - h_{k-1}\|_{\infty} x_k^{n,N}) \\
&\le \exp(-CNnx_k^{n,N}).
\end{aligned}
$$

Finally, since $N_k = \log(|E_{\varepsilon_k}|)$ for all $k \ge 0$, we deduce that

$$
\begin{aligned}
\tilde{\mathbb{P}} \left( \sup_{h \in \mathcal{T}_{K,K'}} \nu_1(h) > \tau \right) &\le |E_{\varepsilon_0}| \exp(-CNnx_0^{n,N}) + \sum_{k=1}^{\infty} (|E_{\varepsilon_k}| + |E_{\varepsilon_{k-1}}|) \exp(-CNnx_k^{n,N}) \\
&\le \exp(N_0 - CNnx_0^{n,N}) + \sum_{k=1}^{\infty} \exp(N_k + N_{k-1} - CNnx_k^{n,N}).
\end{aligned}
$$

We choose $x_0^{n,N}$ and $x_k^{n,N}$, $k \ge 1$ such that,

$$
\begin{aligned}
N_0 - CNnx_0^{n,N} &= -a(K + K' + 2M) - b \\
N_k + N_{k-1} - CNnx_k^{n,N} &= -k(K + K' + 2M) - a(K + K' + 2M) - b
\end{aligned}
$$

where $a$ and $b$ are two positive real numbers. We deduce that

$$x_k^{n,N} \le C_0(1+k) \frac{K+K'+2M}{Nn} \quad \text{and} \quad \tau \le C_1 \sigma_1^2 \sqrt{L} \frac{K+K'+2M}{Nn} \qquad (2.59)$$

with $C_0 > 0$ and $C_1$ two constants depending on $a$ and $b$. It follows that

$$\tilde{\mathbb{P}} \left( \sup_{t \in \mathcal{T}_{K,K'}} \nu(t) > \tau \right) \le \frac{e}{e-1} e^{-b} \exp \left\{ -a \left( K + K' + 2M \right) \right\}.$$

From Equation (2.59), we set

$$q_1(K, K') = \kappa^* \sigma_1^2 \sqrt{L} \frac{K + K' + 2M}{Nn}$$

where $\kappa^* > 0$ depends on $C_1 > 0$. Thus, for all $K, K' \in K$,

$$\mathbb{P} \left( \left\{ \sup_{h \in \mathcal{T}_{K,K'}} \nu^2(h) > q_1(K, K') \right\} \cap \Omega_{n,N,K_{\max}} \right) \le \frac{e^{-b+1}}{e+1} \exp \left\{ -a (K + K' + 2M) \right\}$$

and there exists constants $c, C > 0$ such that

$$
\begin{aligned}
&\mathbb{E} \left[ \left( G_K(K') - q_1(K, K') \right) + 1_{\Omega_{n,N,K_{\max}}} \right] \\
&\le \frac{c(K+K')}{Nn} \mathbb{P} \left( \left\{ \sup_{t \in \mathcal{T}_{K,K'}} \nu^2(t) > q_1(K, K') \right\} \cap \Omega_{n,N,K_{\max}} \right) \\
&\le \frac{C}{Nn} \exp \left\{ -\frac{a}{2} (K+K') \right\}.
\end{aligned}
$$
Finally, there exists a real constant $C > 0$ such that,

$$ \mathbb{E} \left[ \left( G_K(\hat{K}) - q_1(K, \hat{K}) \right)_+ 1_{\Omega_{n,N,K_{\max}}} \right] \le \sum_{K' \in \mathcal{K}} \mathbb{E} \left[ \left( G_K(K') - q_1(K, K') \right)_+ 1_{\Omega_{n,N,K_{\max}}} \right] \le \frac{C}{Nn}. $$

We choose the penalty function $\text{pen}$ such that for each $K \in \mathcal{K}$,

$$ \text{pen}_1(K) \ge \kappa \sigma_1^2 \sqrt{L} \frac{K+M}{Nn}. $$

For $N$ large enough, one has $\sigma_1^2 \le \sqrt{L}$. Thus, we finally set $\text{pen}_1(K) = \kappa \frac{(K+M) \log(N)}{Nn}$ with $L = \log(N)$. Then, there exists a constant $C > 0$ such that,

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_I^2 \right\|_{n,N}^2 \right] \le 34 \inf_{K \in \mathcal{K}} \left\{ \inf_{h \in S_{K+M}} \left\| h - \sigma_I^2 \right\|_n^2 + \text{pen}_1(K) \right\} + \frac{C}{Nn}. $$

□

### Proof of Theorem 2.6.2

*Proof.* From Equation (2.17), we have

$$ \hat{K} := \arg\min_{K \in \mathcal{K}} \gamma_{n,N}(\hat{\sigma}_{K+M,L}^2) + \text{pen}_2(K). $$

Then, for all $K \in \mathcal{K}$ and for all $h \in S_{K+M,L}$, we have

$$ \gamma_{n,N}(\hat{\sigma}_{\hat{K},L}^2) + \text{pen}_2(\hat{K}) \le \gamma_{n,N}(h) + \text{pen}_2(K). $$

Then, for all $K \in \mathcal{K}$ and for all $h \in S_{K+M,L}$,

$$ \gamma_{n,N}(\hat{\sigma}_{\hat{K},L}^2) - \gamma_{n,N}(\sigma^2) \le \gamma_{n,N}(h) - \gamma_{n,N}(\sigma^2) + \text{pen}_2(K) - \text{pen}_2(\hat{K}) \\ \left\| \hat{\sigma}_{\hat{K},L}^2 - \sigma^2 \right\|_{n,N}^2 \le \|h - \sigma^2\|_{n,N}^2 + 2\nu(\hat{\sigma}_{\hat{K},L}^2 - h) + 2\mu(\hat{\sigma}_{\hat{K},L}^2 - h) + \text{pen}_2(K) - \text{pen}_2(\hat{K}). $$

We have for all $a > 0$,

$$ 2\mathbb{E}\left[\mu\left(\hat{\sigma}_{\hat{K},L}^2 - h\right)\right] \le \frac{2}{a}\mathbb{E}\left\|\hat{\sigma}_{\hat{K},L}^2 - \sigma^2\right\|_{n,N}^2 + \frac{2}{a}\inf_{h \in S_{K+M,L}} \|h - \sigma^2\|_n^2 + \frac{a}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} \mathbb{E}\left[(R_{k\Delta}^j)^2\right] $$

and since $\nu = \nu_1 + \nu_2 + \nu_3$, according to the proof of Theorem 2.4.3, there exists a constant $c > 0$ such that

$$ \mathbb{E} \left[ \nu(\hat{\sigma}_{\hat{K},L}^2 - h) \right] \le c \mathbb{E} \left[ \nu_1(\hat{\sigma}_{\hat{K},L}^2 - h) \right] $$

where the for $i \in \{1, 2, 3\}$ and for all $h \in S_{K+M,L}$, $K \in \mathcal{K}$,

$$ \nu_i(h) = \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} h(X_{k\Delta}^j) \zeta_{k\Delta}^j, $$

and the $\zeta_{k\Delta}^j$ are given Then,

$$ \left(1 - \frac{2}{a}\right) \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K},L}^2 - \sigma^2 \right\|_{n,N}^2 \right] \le \left(1 + \frac{2}{a}\right) \inf_{h \in S_{K+M,L}} \|h - \sigma^2\|_n^2 + 2c\mathbb{E} \left[ \nu_1(\hat{\sigma}_{\hat{K},L}^2 - h) \right] \\ + \text{pen}_2(K) - \text{pen}_2(\hat{K}) + \frac{a}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} \mathbb{E} \left[ (R_{k\Delta}^j)^2 \right] $$
From Equation (2.33) and for $a = 4$, there exists a constant $C > 0$ such that

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K},L}^2 - \sigma^2 \right\|_{n,N}^2 \right] \le 3 \inf_{h \in S_{K+M,L}} \left\| h - \sigma^2 \right\|_n^2 + 4c \mathbb{E} \left[ \nu_1(\hat{\sigma}_{\hat{K},L}^2 - h) \right] + 2 \left( \text{pen}_2(K) - \text{pen}_2(\hat{K}) \right) + C \Delta^2. \quad (2.60) $$

Since for all $K \in \mathcal{K}$, $\text{pen}_2(K) \ge 2\kappa^* \sigma_1^2 \sqrt{(K+M)L/(Nn)}$, define the function $q_2: (K, K') \mapsto q_2(K, K')$ such that

$$ q_2(K, K') = 2C^*\sigma_1^2 \sqrt{\frac{(K+K'+2M)L}{Nn}} \ge 2\sigma_1^2 v \sqrt{x^{n,N}} + \sigma_1^2 vx^{n,N} \quad (2.61) $$

where

$$ x^{n,N} \propto \frac{K+K'+2M}{Nn} \quad \text{and} \quad v = \sqrt{2L}. $$

The constant $C^* > 0$ depends on constants $\kappa^* > 0$ and $c > 0$ of Equation (2.60) such that

$$ 4cq_2(K, K') \le \text{pen}_2(K) + 2\text{pen}_2(K'). $$

Then for all $K \in \mathcal{K}$ and for all $h \in S_{K+M,L}$,

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K},L}^2 - \sigma^2 \right\|_{n,N}^2 \right] \le 3 \left( \inf_{h \in S_{K+M,L}} \left\| h - \sigma^2 \right\|_n^2 + \text{pen}_2(K) \right) + 4c \mathbb{E} \left[ \left( \nu_1(\hat{\sigma}_{\hat{m},L}^2 - h) - q_2(K, \hat{K}) \right)_+ \right] + C \Delta^2. \quad (2.62) $$

For all $K \in \mathcal{K}$ and for all $h \in S_{K+M,L}$ such that $\|h\|_{\infty} \le \sqrt{L}$, we have,

$$ \left\| \hat{\sigma}_{\hat{K},L}^2 - h \right\|_{n,N}^2 \le \left\| \hat{\sigma}_{\hat{K},L}^2 - h \right\|_{\infty}^2 \le 2L =: v^2. $$

Then, using Equation (2.61) and Lemma 2.9.10, there exists a constant $C > 0$ such that for all $K, K' \in \mathcal{K}$ and for all $h \in S_{K+M,L}$,

$$ \mathbb{P} \left( \nu_1(\hat{\sigma}_{K',L}^2 - h) \ge q(K, K'), \left\| \hat{\sigma}_{\hat{K},L}^2 - h \right\|_{n,N}^2 \le v^2 \right) \le \exp(-CNnx^{n,N}). \quad (2.63) $$

Since $L = \log(Nn)$, then for $N$ large enough, $\sigma_1^2 \le \sqrt{\log(Nn)}$, we finally choose

$$ \text{pen}_2(K) = \kappa \sqrt{\frac{(K+M) \log^2(Nn)}{Nn}} $$

where $\kappa > 0$ is a new constant. Since $\mathbb{E}[\nu_1(\hat{\sigma}_{\hat{K},L}^2 - h)] \le O\left(\sqrt{\frac{(K_{\max}+M)\log^2(N)}{Nn}}\right)$ (see proof of Theorem 2.4.3), for all $K \in \mathcal{K}$ and $h \in S_{K+M,L}$, there exists a constant $c > 0$ such that

$$ \mathbb{E} \left[ \left( \nu_1(\hat{\sigma}_{\hat{K},L}^2 - h) - q_2(K, \hat{K}) \right)_+ \right] \le \max_{K' \in \mathcal{K}} \left\{ \mathbb{E} \left[ \left( \nu_1(\hat{\sigma}_{K',L}^2 - h) - q_2(K, K') \right)_+ \right] \right\} \\ \le cq_2(K, K_{\max}) \max_{K' \in \mathcal{K}} \left\{ \mathbb{P} \left( \nu_1(\hat{\sigma}_{K',L}^2 - h) \ge q_2(K, K') \right) \right\}. $$

From Equation (2.63), we obtain that

$$ \mathbb{E} \left[ \left( \nu_1(\hat{\sigma}_{\hat{K},L}^2 - h) - q_2(K, \hat{K}) \right)_+ \right] \le cq_2(K, K_{\max}) \exp(-C(K + K_{\max})) \le \frac{C}{\sqrt{Nn}}. \quad (2.64) $$

since $K$ and $K_{\max}$ increase with the size $N$ of the sample paths $D_{N,n}$, and

$$ c\sqrt{Nn}q_2(K, K_{\max}) \exp(-C(K + K_{\max})) \to 0 \quad \text{as} \quad N, n \to \infty \quad (\text{or } N = 1 \text{ and } n \to \infty). $$

Then, from Equations (2.64) and (2.62), there exists a constant $C > 0$ such that

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K},L}^2 - \sigma^2 \right\|_{n,N}^2 \right] \le 3 \inf_{K \in \mathcal{K}} \left\{ \inf_{h \in S_{K+M,L}} \left\| h - \sigma^2 \right\|_n^2 + \text{pen}_2(K) \right\} + \frac{C}{\sqrt{Nn}}. $$

□
### Proof of Theorem 2.6.3

*Proof.* The proof of Theorem 2.6.3 is similar to the proof of Theorem 2.6.1. Then, from Equation (2.34), for all $h \in S_{K+M}$,

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2} \right\|_{n,1}^2 \mathbb{1}_{\Omega_{n,K_{\max}}} \right] \le 3 \inf_{h \in S_{K+M}} \left\| h - \sigma_{|I|^2} \right\|_{n}^2 + 20 \mathbb{E} \left( \sup_{h \in S_{K+M}, \|h\|_X = 1} \nu^2(h) \mathbb{1}_{\Omega_{n,K_{\max}}} \right) + C \Delta^2 + 2 (\mathrm{pen}(K) - \mathrm{pen}(\hat{K})), $$

where $\mathcal{T}_{K,K'} = \{h \in S_{K+M} + S_{K'+M}, \|h\|_X = 1, \|h\|_{\infty} \le \sqrt{L}\}$. Let $q_3: \mathcal{K}^2 \to \mathbb{R}_+$ such that $160q_3(K, K') \le 18\mathrm{pen}_3(K) + 16\mathrm{pen}_3(K')$.

Recall that the $L^2$-norm $\|\cdot\|$, the norm $\mathbb{E}[\|\cdot\|_X]$ and the empirical norm $\|\cdot\|_n$ are equivalent on $L^2(I)$ since the transition density is bounded on the compact interval $I$. Then, for all $K \in \mathcal{K}$ and $h \in S_{K+M,L}$, we have

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2} \right\|_{n,1}^2 \right] \le 3 \left( \inf_{h \in S_{K+M,L}} \left\| h - \sigma_{|I|^2} \right\|_{n}^2 + \mathrm{pen}_3(K) \right) + 20 \mathbb{E} \left[ \sup_{h \in \mathcal{T}_{K,\hat{K}}} \nu_1^2(h) - q_3(K, \hat{K}) \mathbb{1}_{\Omega_{n,K_{\max}}} \right] + C \Delta^2 + CL \frac{K_{\max}^{2\gamma}}{n^{\gamma/2}}, $$

where the numerical constant $\gamma$ can take any value in $(1, +\infty)$, and

$$ \nu_1(h) := \frac{1}{n} \sum_{k=0}^{n-1} h(X_{k\Delta}^1) \zeta_{k\Delta}^{1,1} $$

with $\zeta_{k\Delta}^{1,1}$ the error term. We set for all $K, K' \in \mathcal{K}, G_K(K') := \sup_{h \in \mathcal{T}_{K,K'}} \nu_1^2(h)$ and $\gamma = 20$. Then, for $K_{\max} = n^{1/5}$ and $L = \log(n)$, there exists $C > 0$ such that

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K}}^2 - \sigma_{|I|^2} \right\|_{n,1}^2 \right] \le 3 \inf_{K \in \mathcal{K}} \left\{ \inf_{h \in S_{K+M,L}} \left\| h - \sigma_{|I|^2} \right\|_{n}^2 + \mathrm{pen}_3(K) \right\} + C \Delta^2 + C \frac{\log(n)}{n^2} + 20 \sum_{K' \in \mathcal{K}} \mathbb{E} \left[ (G_K(K') - q_3(K, K')) + \mathbb{1}_{\Omega_{n,K_{\max}}} \right]. $$

Considering the unit ball $\bar{B}_{\|\cdot\|_X}(0,1)$ of the approximation subspace given by

$$ \bar{B}_{\|\cdot\|_X}(0,1) = \{h \in S_{K+M}, \|h\|_X^2 \le 1\} = \{h \in S_{K+M}, \|h\|^2 \le \frac{1}{\tau_0}\}. $$

Then, using a similar reasoning as in Theorem 2.6.1, we obtain that,

$$ \sum_{K' \in \mathcal{K}} \mathbb{E} \left[ (G_K(K') - q_3(K, K'))_+ \right] \le \frac{C}{n}, $$

where $C > 0$ is a constant, $q_3(K, K') \propto \sigma_1^4 \frac{(K+K'+2M)\sqrt{\log(n)}}{n}$ and $\mathrm{pen}_3(K) \propto \frac{(K+M)\log(n)}{n}$. Then we obtain

$$ \mathbb{E} \left[ \left\| \hat{\sigma}_{\hat{K},L}^2 - \sigma_{|I|^2} \right\|_{n,1}^2 \right] \le \frac{3}{\tau_0} \inf_{K \in \mathcal{K}} \left\{ \inf_{h \in S_{K+M,L}} \left\| h - \sigma_{|I|^2} \right\|_{n}^2 + \mathrm{pen}_3(K) \right\} + \frac{C}{n}. $$

□
# Appendix

## Calibration

Fix the drift function $b(x) = 1 - x$, the time-horizon $T = 1$ and at time $t = 0$, $x_0 = 0$. Consider the following three models:

Model 1: $\sigma(x) = 1$

Model 2: $\sigma(x) = 0.1 + 0.9/\sqrt{1 + x^2}$

Model 3: $\sigma(x) = 1/3 + \sin^2(2\pi x)/\pi + 1/(\pi + x^2)$.

The three diffusion models satisfy Assumption 2.2.1 and are used to calibrate the numerical constant $\kappa$ of each penalty function.

As we already know, the adaptive estimator of $\sigma^2$ on the interval $[-\sqrt{\log(N)}, \sqrt{\log(N)}]$ necessitate a data-driven selection of an optimal dimension through the minimization of the penalized least squares contrast given in Equation (2.17). Since the penalty function $\text{pen}(d_N) = \kappa(K_N + M) \log^2(N)/N^2$ depends on the unknown numerical constant $\kappa > 0$, the goal is to select an optimal value of $\kappa$ in the set $\mathcal{V} = \{0.1, 0.5, 1, 2, 4, 5, 7, 10\}$ of its possible values. To this end, we repeat 100 times the following steps:

1. Simulate learning samples $D_N$ and $D_{N'}$ with $N \in \{50, 100\}$, $N' = 100$ and $n \in \{100, 250\}$

2. For each $\kappa \in \mathcal{V}$:

(a) For each $K_N \in \mathcal{K}$ and from $D_N$, compute $\hat{\sigma}_{d_N, L_N}^2$ given in Equations (2.7) and (2.8).

(b) Select the optimal dimension $\hat{K}_N \in \mathcal{K}$ using Equation (2.17)

(c) Using the learning sample $D_{N'}$, evaluate $\|\hat{\sigma}_{d_N, L_N}^2 - \sigma_A^2\|_{n, N'}^2$ where $\hat{d}_N = \hat{K}_N + M$.

Then, we calculate average values of $\|\hat{\sigma}_{d_N, L_N}^2 - \sigma_A^2\|_{n, N'}^2$ for each $\kappa \in \mathcal{V}$ and obtain the following results:

<figure><img src="image_11.png" /><figcaption>Figure 2: Calibration of the constant $\kappa \in \mathcal{V}$ of the penalty function</figcaption></figure>

We finally choose $5 \in \mathcal{V}$ as the optimal value of $\kappa$ in reference to the results of Figure 2.
**Proof of Lemma 2.3.1.** The proof is divided into two parts for each of the two results to be proven.

**First result.** Since the function $h$ is continuous on $\mathbb{R}$, let $H$ be a primitive of $h$ on $\mathbb{R}$. We deduce that for all $s \in [0, 1]$,

$$h(X_s) = \lim_{\epsilon \to 0} \frac{H(X_s + \epsilon) - H(X_s - \epsilon)}{2\epsilon} = \lim_{\epsilon \to 0} \frac{1}{2\epsilon} \int_{X_s - \epsilon}^{X_s + \epsilon} h(x)dx = \lim_{\epsilon \to 0} \frac{1}{2\epsilon} \int_{-\infty}^{+\infty} h(x)1_{(x-\epsilon, x+\epsilon)}(X_s)dx.$$

Finally, since $h$ is integrable on $\mathbb{R}$ and using the theorem of dominated convergence, we obtain

$$\int_{0}^{1} h(X_s)ds = \int_{-\infty}^{+\infty} h(x)\lim_{\epsilon \to 0} \frac{1}{2\epsilon} \int_{0}^{1} 1_{(x-\epsilon, x+\epsilon)}(X_s)dsdx = \int_{-\infty}^{+\infty} h(x)\mathcal{L}^x dx.$$

**Second result.** Fix $t \in (0, 1]$ and consider $P_X : (t, x) \mapsto \int_{-\infty}^{x} p_X(t, y)dy$ the cumulative density function of the random variable $X_t$ of the density function $x \mapsto p_X(t, x)$. We have:

$$\begin{aligned}
\forall x \in \mathbb{R}, \quad \mathbb{E}(\mathcal{L}^x) &= \lim_{\epsilon \to 0} \frac{1}{2\epsilon} \int_0^1 \mathbb{E}[1_{(x-\epsilon, x+\epsilon)}(X_s)] ds = \lim_{\epsilon \to 0} \frac{1}{2\epsilon} \int_0^1 \mathbb{P}(x-\epsilon \le X_s \le x+\epsilon) ds \\
&= \int_0^1 \lim_{\epsilon \to 0} \frac{P_X(s, x+\epsilon) - P_X(s, x-\epsilon)}{2\epsilon} ds \\
&= \int_0^1 p_X(s, x) ds.
\end{aligned}$$

□

**Proof of Lemma 2.9.5.** From [86], Theorem 1.7, we have

$$\forall x \in \mathbb{R}, \quad \mathcal{L}^x - \mathcal{L}^{x-} = 2 \int_0^1 1_{X_s=x} dX_s = 2 \int_0^1 1_{X_s=x} b(X_s) ds + 2 \int_0^1 1_{X_s=x} \sigma(X_s) dW_s.$$

For each $s \in [0, 1]$, let $F_s$ be the cumulative density function of the random variable $X_s$. For all $x \in \mathbb{R}$ and for all $s \in [0, 1]$, we have for all $\epsilon > 0$,

$$\begin{aligned}
\mathbb{P}(X_s = x) &= \lim_{\epsilon \to 0} \mathbb{P}(X_s \le x + \epsilon) - \lim_{\epsilon \to 0} \mathbb{P}(X_s \le x - \epsilon) = \lim_{\epsilon \to 0} F_s(x + \epsilon) - \lim_{\epsilon \to 0} F_s(x - \epsilon) \\
&= F_s(x) - F_s(x^{-}) \\
&= 0
\end{aligned}$$

Thus, for all $x \in \mathbb{R}$,

$$\begin{aligned}
\mathbb{E}[|\mathcal{L}^x - \mathcal{L}^{x-}|] &\le 2 \int_0^1 |b(x)| \mathbb{P}(X_s = x) ds + 2 \mathbb{E} \left[ \left| \int_0^1 1_{X_s=x} \sigma(X_s) dW_s \right| \right] \\
&= 2 \mathbb{E} \left[ \left| \int_0^1 1_{X_s=x} \sigma(X_s) dW_s \right| \right].
\end{aligned}$$

Using the Cauchy Schwartz inequality, we conclude that

$$\mathbb{E}[|\mathcal{L}^x - \mathcal{L}^{x-}|] \le 2\sqrt{\mathbb{E}\left(\int_0^1 1_{X_s=x} \sigma^2(X_s) ds\right)} = 2\sigma(x) \int_0^1 \mathbb{P}(X_s = x) ds = 0.$$

Using the Markov inequality, we have

$$\forall \epsilon > 0, \quad \mathbb{P}(|\mathcal{L}^x - \mathcal{L}^{x-}| > \epsilon) \le \frac{1}{\epsilon} \mathbb{E}[|\mathcal{L}^x - \mathcal{L}^{x-}|] = 0.$$

We finally conclude that for all $x \in \mathbb{R}$,

$$\mathbb{P}(\mathcal{L}^x \ne \mathcal{L}^{x-}) = \mathbb{P}(|\mathcal{L}^x - \mathcal{L}^{x-}| > 0) = 0.$$

□
**Proof of Lemma 2.9.6.** The proof of this Lemma mainly focuses on the spline basis and the Fourier basis based on functions cos and sin which are Lipschitz functions. Thus, for all $g = \sum_{\ell=0}^{m-1} a_\ell \phi_\ell \in S_m$,

$$ \|g\|_{n,1}^2 - \|g\|_X^2 \le \int_0^1 |g^2(X_{\eta(s)}) - g^2(X_s)| ds \le 2\|g\|_{\infty} \int_0^1 |g(X_{\eta(s)}) - g(X_s)| ds. \quad (65) $$

From Equation (2.11), one has $\mathbb{E}[\|g\|_X^2] \ge \tau_0 \|g\|^2$. Thus, if $\|g\|_X^2 = 1$, then $\|g\|^2 \le 1/\tau_0$, and we deduce for all $g = \sum_{\ell=0}^{m-1} a_\ell \phi_\ell$ that there exists a constant $C > 0$ such that

* Collection [B]: $\|g\|_{\infty} \le \|a\|_2 \le C\sqrt{m}$ (see [30])

* Collection [CS-OB]: $\|g\|_{\infty} \le Cm^{r/2}$ since $\|g\| = \|a\|_2$ and $\sum_{\ell=0}^{m-1} \phi_{\ell}^2 = O(m^r)$.

Moreover, each $g \in S_m$ such that $\|g\|_X^2 = 1$ is a Lipschitz function with a Lipschitz coefficient $L_g = O(m^{3/2})$ for the collection [B], or $L_g = O(m^{r/2})$ for the collection [CS-OB]. For the spline basis, this result is obtained in [30], proof of Lemma C.1 combined with Lemma 2.6. For the collection [CS-OB], for all $x, y \in I$ and using the Cauchy-Schwarz inequality together with Equation (2.5) and the finite increments theorem, there exists a constant $C > 0$ such that

$$ \begin{aligned} |g(x) - g(y)| &\le \sum_{\ell=0}^{m-1} |a_\ell| \cdot |\phi_\ell(x) - \phi_\ell(y)| \\ &\le \|a\|_2 \sqrt{\sum_{\ell=0}^{m-1} \|\phi_\ell'\|_{\infty}^2} |x - y| \\ &\le Cm^{r/2} |x - y|. \end{aligned} $$

Back to Equation (65), there exists a constant $C > 0$ such that

$$ \|g\|_{n,1}^2 - \|g\|_X^2 \le Cm^{\alpha} \int_0^1 |X_{\eta(s)} - X_s| ds \quad (66) $$

with $\alpha = 2$ for the collection [B], and $\alpha = (r + r')/2$ for the collection [CS-OB]. We have:

$$ \Omega_{n,m}^c = \left\{ \omega \in \Omega, \exists g \in S_m \setminus \{0\}, \left| \frac{\|g\|_{n,1}^2}{\|g\|_X^2} - 1 \right| > \frac{1}{2} \right\}, $$

and, using Equation (66), we obtain

$$ \sup_{g \in S_m \setminus \{0\}} \left| \frac{\|g\|_{n,1}^2}{\|g\|_X^2} - 1 \right| = \sup_{g \in S_m, \|g\|_X^2=1} \left| \|g\|_{n,1}^2 - \|g\|_X^2 \right| \le Cm^{\alpha} \int_0^1 |X_{\eta(s)} - X_s| ds. $$

Finally, using the Markov inequality, the Hölder inequality, Lemma (2.2.2), and Lemma 2.9.1, we conclude that

$$ \begin{aligned} \mathbb{P}(\Omega_{n,m}^c) &\le \mathbb{P}\left(Cm^{\alpha} \int_0^1 |X_{\eta(s)} - X_s| ds \ge \frac{1}{2}\right) \\ &\le Cm^{\alpha\gamma} \int_0^1 \mathbb{E}[|X_{\eta(s)} - X_s|^\gamma] ds \\ &\le C \frac{m^{\alpha\gamma}}{n^{\gamma/2}} \end{aligned} $$

with $\gamma \in (1, +\infty)$.

**Proof of Lemma 2.9.8.** We have:

$$ \Omega_{n,N,m}^c = \left\{ \omega \in \Omega, \exists h_0 \in S_m, \left| \frac{\|h\|_{n,N}^2}{\|h\|_n^2} - 1 \right| > \frac{1}{2} \right\}, $$
Denote by $\mathcal{H}_m = \{h \in S_m, \|h\|_n = 1\}$. We have

$$ \sup_{h \in \mathcal{H}_m} \left| \frac{\|h\|_{n,N}^2}{\|h\|_n^2} - 1 \right| = \sup_{h \in \mathcal{H}_m} \left| \|h\|_{n,N}^2 - 1 \right|. $$

Let $\varepsilon > 0$ and let $\mathcal{H}_m^\varepsilon$ be the $\varepsilon$-net of $\mathcal{H}_m$ w.r.t. the supremum norm $\|\cdot\|_\infty$. Then, for each $h \in \mathcal{H}_m$, there exists $h_\varepsilon \in \mathcal{H}_m^\varepsilon$ such that $\|h - h_\varepsilon\|_\infty \le \varepsilon$. Then

$$ \|h\|_{n,N}^2 - 1 \le \|h\|_{n,N}^2 - \|h_\varepsilon\|_{n,N}^2 + \|h_\varepsilon\|_{n,N}^2 - 1 $$

and,

$$ \|h\|_{n,N}^2 - \|h_\varepsilon\|_{n,N}^2 \le \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} \left| h(X_{k\Delta}^j) - h_\varepsilon(X_{k\Delta}^j) \right| (\|h\|_\infty + \|h_\varepsilon\|_\infty) \le (\|h\|_\infty + \|h_\varepsilon\|_\infty) \varepsilon. $$

Moreover, we have $\|h\|^2, \|h_\varepsilon\|^2 \le 1/\tau_0$. Then, there exists a constant $c > 0$ such that

$$ \begin{cases} \|h\|_{n,N}^2 - \|h_\varepsilon\|_{n,N}^2 \le 2\sqrt{\frac{cm}{\tau_0}}\varepsilon & \text{for the spline basis (see Lemma 2.6 in Denis et al. (2021))} \\ \|h\|_{n,N}^2 - \|h_\varepsilon\|_{n,N}^2 \le 2\sqrt{\frac{cm}{\tau_0}}\varepsilon & \text{for an orthonormal basis } (\|h\|_\infty^2 \le (\max_{0 \le l \le m-1} \|\phi_l\|_\infty^2)m\|h\|^2). \end{cases} $$

Therefore, for all $\delta > 0$ and for both the spline basis and any orthonormal basis,

$$ \mathbb{P} \left( \sup_{h \in \mathcal{H}_m} \left| \|h\|_{n,N}^2 - 1 \right| \ge \delta \right) \le \mathbb{P} \left( \sup_{h \in \mathcal{H}_m^\varepsilon} \left| \|h\|_{n,N}^2 - 1 \right| \ge \delta/2 \right) + 1_{4\varepsilon\sqrt{\frac{cm}{\tau_0}} \ge \delta}. $$

We set $\delta = 1/2$ and we choose $\varepsilon > 0$ such that $4\varepsilon\sqrt{\frac{cm}{\tau_0}} < 1/2$. Then, using the Hoeffding inequality, there exists a constant $c > 0$ depending on $c$ and $\tau_0$ such that

$$ \mathbb{P} (\Omega_{n,N,m}^c) \le 2\mathcal{N}_\infty(\varepsilon, \mathcal{H}_m) \exp \left( -c \frac{N}{m} \right) \quad (67) $$

where $\mathcal{N}_\infty(\varepsilon, \mathcal{H}_m)$ is the covering number of $\mathcal{H}_m$ satisfying:

$$ \mathcal{N}_\infty(\varepsilon, \mathcal{H}_m) \le \left( \kappa \frac{\sqrt{m}}{\varepsilon} \right)^m \quad (68) $$

where the constant $\kappa > 0$ depends on $c > 0$ (see [30], Proof of Lemma D.1). We set $\varepsilon = \frac{\kappa\sqrt{m^*}}{N}$ with $m^* = \max \mathcal{M}$ and we derive from Equations (67) and (68) that

$$ \mathbb{P} (\Omega_{n,N,m}^c) \le 2N^{m^*} \exp \left( -c \frac{N}{m^*} \right) = 2 \exp \left( -c \frac{N}{m^*} \left( 1 - \frac{m^{*2} \log(N)}{cN} \right) \right). $$

* If $n \ge N$, then $m \in \mathcal{M} = \{1, \dots, \sqrt{N}/\log(Nn)\}$. Since $m^{*2}\log(N)/N \to 0$ as $N \to +\infty$, there exists a constant $C > 0$ such that

$$ \mathbb{P} (\Omega_{n,N,m}^c) \le 2 \exp (-C\sqrt{N}). $$

* If $n \le N$, then $m \in \mathcal{M} = \{1, \dots, \sqrt{n}/\log(Nn)\}$, $m^{*2}\log(N)/N \le \log(N)/\log^2(Nn) \to 0$ as $N, n \to \infty$, and

$$ \mathbb{P} (\Omega_{n,N,m}^c) \le 2 \exp (-C\sqrt{n}). $$

* If $n \propto N$, then $m \in \mathcal{M} = \{1, \dots, \sqrt{N}/\log(Nn)\}$. Since $m^{*2}\log(N)/N \to 0$ as $N \to +\infty$, there exists a constant $C > 0$ such that

$$ \mathbb{P} (\Omega_{n,N,m}^c) \le 2 \exp (-C\sqrt{N}). $$

□
**Proof of Lemma 2.9.10**

*Proof.* We obtain from Comte, Genon-Catalot, Rozenholc (2007) proof of Lemma 3 that for each $j \in [1, N]$, $k \in [0, n-1]$ and $p \in \mathbb{N} \setminus \{0, 1\}$

$$ \mathbb{E} \left[ \exp \left( ug(X_{k\Delta}^j) \xi_{k\Delta}^{j,1} - \frac{au^2 g^2(X_{k\Delta}^j)}{1 - bu} \right) | \mathcal{F}_{k\Delta} \right] \le 1 $$

with $a = e(4\sigma_1^2 c^2)^2$, $b = 4\sigma_1^2 c^2 e \|g\|_\infty$, $u \in \mathbb{R}$ such that $bu < 1$ and $c > 0$ a real constant. Thus,

$$ \begin{aligned} \mathbb{P} \left( \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} g(X_{k\Delta}^j) \xi_{k\Delta}^{j,1} \ge \varepsilon, \|g\|_{N,n}^2 \le v^2 \right) &= \mathbb{E} \left( \mathbf{1}_{\{\sum_{j=1}^{N} \sum_{k=0}^{n-1} ug(X_{k\Delta}^j) \xi_{k\Delta}^{(j,1)} \ge Nnu\varepsilon\}} \mathbf{1}_{\|g\|_{n,N}^2 \le v^2} \right) \\ &= \mathbb{E} \left( \mathbf{1}_{\{\exp(\sum_{j=1}^{N} \sum_{k=0}^{n-1} ug(X_{k\Delta}^j) \xi_{k\Delta}^{(j,1)}) e^{-Nnu\varepsilon} \ge 1\}} \mathbf{1}_{\|g\|_{N,n}^2 \le v^2} \right) \\ &\le e^{-Nnu\varepsilon} \mathbb{E} \left[ \mathbf{1}_{\|g\|_{n,N}^2 \le v^2} \exp \left\{ \sum_{j=1}^{N} \sum_{k=0}^{n-1} ug(X_{k\Delta}^j) \xi_{k\Delta}^{j,1} \right\} \right]. \end{aligned} $$

It follows that,

$$ \mathbb{P} \left( \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} g(X_{k\Delta}^j) \xi_{k\Delta}^{j,1} \ge \varepsilon, \|g\|_{n,N}^2 \le v^2 \right) \le \exp \left\{ -Nnu\varepsilon + \frac{Nnau^2v^2}{1-bu} \right\}. $$

We set $u = \frac{\varepsilon}{\varepsilon b + 2av^2}$. Then, we have $-Nnu\varepsilon + Nnav^2u^2/(1-bu) = -Nn\varepsilon^2/2(\varepsilon b + 2av^2)$ and,

$$ \begin{aligned} \mathbb{P} \left( \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} g(X_{k\Delta}^j) \xi_{k\Delta}^{j,1} \ge \varepsilon, \|g\|_{n,N}^2 \le v^2 \right) &\le \exp \left( - \frac{Nn\varepsilon^2}{2(\varepsilon b + av^2)} \right) \\ &\le \exp \left( -C \frac{Nn\varepsilon^2}{\sigma_1^2 (\varepsilon \|g\|_\infty + 4\sigma_1^2 v^2)} \right) \end{aligned} $$

where $C > 0$ is a constant depending on $c > 0$. $\square$

**Proof of Lemma 2.9.9**

*Proof.* Set $K_{n,N} = K_N$ since $N \propto n$. Let us remind the reader of the Gram matrix $\Psi_{K_N}$ given in Equation (2.14),

$$ \Psi_{K_N} = \mathbb{E} \left[ \frac{1}{Nn} \mathbf{F}'_{K_N} \mathbf{F}_{K_N} \right] = \mathbb{E} \left( \hat{\Psi}_{K_N} \right) $$

where,

$$ \mathbf{F}_{K_N} := \left( (B_\ell(X_0^j), \dots, (B_\ell(X_{(n-1)\Delta}^j)) \right)_{\substack{0 \le \ell \le K_N-1 \\ 1 \le j \le N}} \in \mathbb{R}^{Nn \times (K_N+M)} $$

The empirical counterpart $\hat{\Psi}$ is the random matrix given by $\hat{\Psi}_{K_N}$ of size $(K_N + M) \times (K_N + M)$ is given by

$$ \hat{\Psi}_{K_N} := \frac{1}{Nn} \mathbf{F}'_{K_N} \mathbf{F}_{K_N} = \left( \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} f_\ell(X_{k\Delta}^j) f_{\ell'}(X_{k\Delta}^j) \right)_{\ell, \ell' \in [-M, K_N-1]} . $$

For all $t = \sum_{\ell=-M}^{K_N-1} a_\ell B_{\ell,M,u} \in S_{K_N,M}$ one has

$$ \|t\|_{n,N}^2 = a' \hat{\Psi}_{K_N} a \quad \text{and} \quad \|t\|_n^2 = a' \Psi_{K_N} a, \quad \text{with} \quad a = (a_{-M}, \dots, a_{K_N-1})'. $$
Under Assumption 2.2.1, we follow the lines of [21] Proposition 2.3 and Lemma 6.2. Then,

$$
\begin{align*}
\sup_{t \in S_{K_N, M}, \|t\|_n = 1} \|t\|_{n,N}^2 - \|t\|_n^2 &= \sup_{w \in \mathbb{R}^{K_N+M}, \|\Phi_{K_N}^{1/2} w\|_{2,K_N+M} = 1} \left| w' (\widehat{\Psi}_{K_N} - \Psi_{K_N}) w \right| \\
&= \sup_{u \in \mathbb{R}^{K_N+M}, \|u\|_{2,K_N+M} = 1} \left| u' \Psi_{K_N}^{-1/2} (\widehat{\Psi}_{K_N} - \Psi_{K_N}) \Psi_{K_N}^{-1/2} u \right| \\
&= \left\| \Psi_{K_N}^{-1/2} \widehat{\Psi}_{K_N} \Psi_{K_N}^{-1/2} - \mathrm{Id}_{K_N+M} \right\|_{\mathrm{op}}.
\end{align*}
$$

Therefore,

$$
\Omega_{n,N,K_N}^c = \left\{ \left\| \Psi_{K_N}^{-1/2} \widehat{\Psi}_{K_N} \Psi_{K_N}^{-1/2} - \mathrm{Id}_{K_N+M} \right\|_{\mathrm{op}} > \frac{1}{2} \right\} .
$$

Since $A_N = o(\sqrt{\log(N)})$, we obtain from [28], *proof of Lemma 7.8*, there exists a constant $C > 0$
such that

$$
\mathbb{P} (\Omega_{n,N,K_N}^c) \le 2(K_N + M) \exp (-C \log^{3/2}(N)). \quad (69)
$$

Finally, since $2(K_N + M) \exp(-(C/2)\log^{3/2}(N)) \to 0$ as $N \to +\infty$, one concludes from Equation (69) and for $N$ large enough,

$$
\mathbb{P} (\Omega_{n,N,K_N}^c) \leq C \exp (-c \log^{3/2}(N))
$$

where $c > 0$ and $C > 0$ are new constants.

<figure><img src="image_12.png" /></figure>
# Procédure non-paramétrique de classification multiclases de type plug-in pour les trajectoires de diffusions

Résumé Nous étudions le problème de classification multiclases où la variable vient d'un mélange de diffusions homogènes en temps. Spécifiquement, les classes sont discriminées par leurs fonctions de dérive tandis que le coefficient de diffusion est commun à toutes les classes et inconnu. Dans ce cadre, nous construisons un classifieur de type plug-in qui repose sur des estimateurs non-paramétriques des fonctions de dérive et de diffusion. Nous établissons premièrement la consistance de notre procédure de classification sous des hypothèses faibles sur le modèle étudié, et ensuite, nous fournissons des vitesses de convergence sous des ensembles d'hypothèses différents. Finallement, une étude numérique soutient nos résultats théoriques.

**Mots clés:** Apprentissage supervisé; Classification multiclases; Estimation non-paramétrique; Classifieur de type plug-in; Processus de diffusion

**Abstract:** We study the multiclass classification problem where the features come from a mixture of time-homogeneous diffusions. Specifically, the classes are discriminated by their drift functions while the diffusion coefficient is common to all classes and unknown. In this framework, we build a plug-in classifier which relies on nonparamateric estimators of the drift and diffusion functions. We first establish the consistency of our classification procedure under mild assumptions and then provide rates of convergence under different set of assumptions. Finally, a numerical study supports our theoretical findings.

**Keywords:** Supervised learning; Multiclass classification; Nonparametric estimation; Plug-in classifier; Diffusion process

## Contents

<table><tr><td>A.1 Introduction</td><td>88</td></tr><tr><td>A.2 Statistical setting</td><td>90</td></tr><tr><td>A.2.1 Assumptions</td><td>90</td></tr><tr><td>A.2.2 Bayes Classifier</td><td>91</td></tr><tr><td>A.3 Classification procedure: a plug-in approach</td><td>91</td></tr><tr><td>A.3.1 Classifier and excess risk</td><td>92</td></tr><tr><td>A.3.2 Estimators of drift and diffusion coefficients</td><td>93</td></tr><tr><td>A.3.3 A general consistency result</td><td>94</td></tr><tr><td>A.3.4 General rate of convergence for bounded drift function</td><td>95</td></tr></table>
<table><tr><td><b>A.4</b></td><td><b>Classifier's rate of convergence with known diffusion coefficient</b></td><td><b>96</b></td></tr><tr><td>A.4.1</td><td>Rates of convergence for drift estimators</td><td>97</td></tr><tr><td>A.4.2</td><td>Rates of convergence: bounded drift functions</td><td>98</td></tr><tr><td>A.4.3</td><td>Rates of convergence: when the drift functions are re-entrant</td><td>99</td></tr><tr><td><b>A.5</b></td><td><b>Simulation study</b></td><td><b>100</b></td></tr><tr><td>A.5.1</td><td>Models and simulation setting</td><td>100</td></tr><tr><td>A.5.2</td><td>Implementation of the plug-in classifier</td><td>101</td></tr><tr><td>A.5.3</td><td>Simulation results</td><td>102</td></tr><tr><td>A.5.4</td><td>Ornstein-Uhlenbeck model</td><td>102</td></tr><tr><td><b>A.6</b></td><td><b>Conclusion and discussion</b></td><td><b>104</b></td></tr><tr><td><b>A.7</b></td><td><b>Proofs</b></td><td><b>104</b></td></tr><tr><td>A.7.1</td><td>Technical results on the process X</td><td>104</td></tr><tr><td>A.7.2</td><td>Proofs of Section A.3</td><td>108</td></tr><tr><td>A.7.3</td><td>Proofs of Section A.4</td><td>118</td></tr><tr><td><b>A.8</b></td><td><b>Appendix</b></td><td><b>126</b></td></tr></table>

## A.1 Introduction

The massive collection of functional data has found many applications in recent years for the modeling of the joint (time)-evolution of agents – individuals, species, particles – that are represented by some sets of features – time-varying variables such as geographical positions, population sizes, portfolio values etc. Examples can be found in mathematical finance (see e.g. [35]), biology (see e.g. [37]), or physics (see e.g. [33]). This gave rise to an abundant literature on statistical methods for functional data, (see e.g. [84, 99], for a review). Within this context, the study of efficient supervised classification procedures that are designed to handle temporal data is a major challenge. Indeed, usual learning algorithms such as random forests, kernel methods or neural networks are not directly tailored to take into account the temporal dependency of the data. Recently, this question has drawn a lot of attention, see [24, 5, 100, 26, 68] any references therein.

In the present paper, we tackle the multiclass classification problem where the features belong to a particular family of functional data, namely trajectories, whose temporal dynamic is modelled by stochastic differential equation. In this framework, we propose a nonparametric plug-in type procedure for such data generated by diffusion processes observed at discrete time. Hence, our work takes place in the high frequency setup. Let us denote by $(X, Y)$ a random couple built on a probability space $(\Omega, \mathcal{F}, \mathbb{P}_{(X,Y)})$. The feature $X = (X_t)_{t \in [0,1]}$ is a real-valued diffusion process whose drift coefficient depends on its associated label $Y$ taking values in $\mathcal{Y} = \{1, \dots, K\}$, with $K \ge 2$. More precisely, for each $i \in \mathcal{Y}$, $X$ is a solution of a stochastic differential equation whose drift function, denoted by $b_i^*$, depends on the class $i$. The marginal distribution of $X$ is hence a mixture of distributions of time-homogeneous diffusion processes. We assume that a learning sample $\mathcal{D}_N = \{(X_t^i)_{t \in [0,1]}, Y_i\}$, $i = 1, \dots, N$ is provided, composed of $N$ i.i.d. random couples with distribution $\mathbb{P}_{(X,Y)}$. Additionally, in this paper, the diffusions $X_i$ are observed on a subdivision $\{0, 1/n, \dots, 1\}$ of the time interval $[0, 1]$, for a positive integer $n$. Since we deal with multiclass classification setting, the statistical goal is then to build, based on $\mathcal{D}_N$, a classifier $\hat{g}$, such that $\hat{g}(X)$ is a prediction of the associated label $Y$ of a new path $X$. Besides, we expect that the empirical classifier mimics the optimal Bayes classifier $g^*$ characterized as

$$g^*(X) \in \arg \min_{g} \mathbb{P}_{(X,Y)} (g(X) \neq Y).$$

Specifically, we propose a classification procedure based on the plug-in principle. In particular, the construction of our empirical classifier relies on estimators of both drift and diffusion coefficients. The performance of a predictor $\hat{g}$ is assessed through its excess risk $\mathbb{P}(\hat{g}(X) \neq Y) - \mathbb{P}(g^*(X) \neq Y)$. In the finite dimensional classification setup (e.g. $X \in \mathbb{R}^d$), rates of convergence for plug-in rules are
usually obtained under the strong density assumption (X admits a density which is lower bounded) as in [4, 48]. However, theoretical properties of plug-in rules in supervised classification of trajectories are much less studied.

**Related works.** Up to our knowledge, the work of [11] is the first one that tackles the problem of supervised classification in the stochastic differential equation framework. More precisely, the authors consider the model where $X = (X_t)_{t \in [0,1]}$ is a mixture of two diffusion processes and provide a classifier based on the empirical risk minimization strategy for which they establish rates of convergence. However, the proposed method is not implementable since it involves the minimization of a non-convex criterion. More recently, [47], and [29] study plug-in classifiers for classification of diffusion paths. In [47] the authors propose a plug-in rule for the binary classification problem where the trajectories are generated by Gaussian processes, solutions of the white noise model. In this model, the drift function depends on time and on the label $Y$, also, the diffusion coefficient is supposed to be constant and known. Within this framework, [47] establish the optimality of their classification procedure which reaches the minimax rate of convergence of order $N^{-s/(2s+1)}$, where the drift function is assumed to belong to a Sobolev space of regularity $s \ge 1$. Under an additional margin type assumption, they also derive faster rates of convergence. Closest to our framework, [29] also consider the challenging multiclass problem where the drift functions are space-dependent. However, the authors consider drift functions modeled under parametric assumptions, keeping the diffusion coefficient known and constant. They propose a plug-in classifier for which only consistency is established.

In the present work, we consider a plug-in classifier that relies on nonparametric estimators of the drift and diffusion coefficients. The literature on this topic is extensive. Usually, the construction of estimators of drift and diffusion functions relies on the observation of a single path. For instance, [61] studies minimax rate of convergence for the estimation of the diffusion coefficient on a compact interval. For the inference of the drift coefficient, the main references using penalized contrasts can be found for long time observation with high frequency data in [60, 22, 20]. However, since we deal with the multiclass classification framework, the construction of estimators of both drift and diffusion coefficients is based on the learning sample $D_N$ which is composed of repeated observations of the process on the fixed time-interval $[0, 1]$. Recently, [18, 75, 27] consider nonparametric procedures for the estimation of the drift function for continuous observations in the context of i.i.d. observations when the horizon time is fixed. Furthermore, towards high-frequency data, [30] study minimum contrast estimator under a $l_2$ constraint.

**Main contributions.** In this paper, we extend the results of [29] and [47] in several directions. In particular, one of the major contribution is to provide, up to our knowledge, the first study of rates of convergence for plug-in classifier in the mixture model of time-homogeneous diffusion. Importantly, we highlight that extending the results of [47] to diffusion models in which the drift functions are space-dependent and the diffusion coefficient is either unknown or non-constant add many difficulties. Besides, contrary to [29], we consider the nonparametric mixture model where both drift and diffusion functions are unknown as well as the weights of the mixture. Specifically, we build a plug-in classifier that relies on the Girsanov's theorem and involves nonparametric estimators of the drift functions $b_i^*, i \in \mathcal{Y}$, and the diffusion coefficient. The construction of our estimators is inspired of the ridge estimators provided in [30], and consists in the minimization of a least-squares type contrast over a finite dimensional subspace under a $l_2$-constraint. The considered space of approximation is then spanned by the $B$-spline basis [25].

One of the main difficulty of the study of statistical properties of the plug-in classifiers in our context is that it requires deriving rates of convergence for the drift and diffusion coefficients on a non-compact interval. It hence implies that the strong density assumption does not hold, although, we consider assumptions that ensure existence of transition density. Notably, our results embed generalization of the results provided in [30] for the estimation of non-compactly supported drift functions for $B$-spline based estimators, but also exhibit the first result for the estimation of the diffusion coefficient in the i.i.d. framework. A salient point of our theoretical findings is obtained when the diffusion
coefficient is constant and known. In this case, by leveraging the results of [18], we show that optimal rates for drift estimation can only be achieved on intervals included in $[-C\sqrt{\log(N)}, C\sqrt{\log(N)}]$, with $C > 0$.

To sum up our results, a first part is dedicated to the consistency of our plug-in classifier which is obtained under very mild assumptions. In a second part, convergence rates are established in three particular cases.

(i) When the drift functions are bounded and Lipschitz, and the diffusion coefficient is unknown and possibly non-constant, we obtain a rate of convergence of order $N^{-1/5}$ for the plug-in classifier (up to a factor of order $\exp(\sqrt{c \log(N)}), c > 0$).

(ii) When the diffusion coefficient is known and constant, and when the drift functions are bounded and belongs to some Hölder space with regularity $\beta$, using some arguments developed in [19] and [20] for the estimation of non-compactly supported drift functions, together with approximations of the transition density of $X$ (as they are intractable), we then prove that the plug-in classifier reaches rate of order $N^{-\beta/(2\beta+1)}$ (up to a factor of order $\exp(\sqrt{c \log(N)}), c > 0$).

(iii) When the drifts are unbounded but re-entrant and Hölder continuous with regularity $\beta$, we obtain a rate of convergence of order $N^{-3\beta/(4(2\beta+1))}$. Notice that when $\beta = 1$ and $d = 1$, it corresponds to the rate found in [48].

## A.2 Statistical setting

We consider the multiclass classification problem, where the feature $X$ comes from a mixture of Brownian diffusions with drift. More precisely, the generic data-structure is a couple $(X, Y)$ where the label $Y$ takes its values in the set $\mathcal{Y} := \{1, \dots, K\}$ with distribution denoted by $\mathfrak{p}^* = (\mathfrak{p}_1^*, \dots, \mathfrak{p}_K^*)$, and where the process $X = (X_t)_{t \in [0,1]}$ is defined as the solution of the following stochastic differential equation

$$dX_t = b_Y^*(X_t)dt + \sigma^*(X_t)dW_t, \quad X_0 = 0, \qquad (A.1)$$

where $(W_t)_{t \ge 0}$ is a standard Brownian motion independent of $Y$. In the following, we denote by $\mathfrak{b}^* = (b_1^*, \dots, b_K^*)$ the vector of drift functions. The real-valued functions $b_i^*(.)$, $i \in \mathcal{Y}$, and the diffusion coefficient $\sigma^*(.)$ are assumed to be unknown. We also assume that $0 < \mathfrak{p}_0^* = \min_{i \in \mathcal{Y}} \mathfrak{p}_i^*.$

In this framework, the objective is to build a classifier $g$, i.e. a measurable function such that the value $g(X)$ is a prediction of the associated label $Y$ of $X$. The accuracy of such classifier $g$ is then assessed through its misclassification risk, denoted by

$$\mathcal{R}(g) := \mathbb{P}_{(X,Y)} (g(X) \neq Y).$$

In the following, the set of all classifiers is denoted by $\mathcal{G}$.

The main assumptions considered throughout the paper are presented in Section A.2.1. The definition and characterization of the optimal classifier w.r.t. the misclassification risk, namely the Bayes classifier, is provided in Section A.2.2

### A.2.1 Assumptions

The following assumptions ensure that Equation (A.1) admits a unique strong solution (see [67], Theorem 2.9), and that the diffusion process $X$ admits a transition density

$$p_X : (t, x) \in ([0, 1] \times \mathbb{R}) \mapsto p_X(t, x)$$

(see for example [55]).

**Assumption A.2.1. (Ellipticity and regularity)**

(i) There exists $L_0 > 0$ such that the functions $b_i^*$, $i = 1, \dots, K$ and $\sigma^*$ are $L_0$-Lipschitz:

$$\sup_{i \in \mathcal{Y}} |b_i^*(x) - b_i^*(y)| + |\sigma^*(x) - \sigma^*(y)| \le L_0 |x - y|, \forall (x, y) \in \mathbb{R}^2.$$
(ii) There exist real constants $\sigma_0^*, \sigma_1^*$ such that

$$0 < \sigma_0^* \le \sigma^*(x) \le \sigma_1^*, \forall x \in \mathbb{R}.$$

(iii) $\sigma^* \in \mathbb{C}^2(\mathbb{R})$ and there exist $\gamma \ge 0$ and $c > 0$ such that: $|\sigma^{*l}(x)| + |\sigma^{*l}(x)| \le c(1 + |x|^\gamma), \forall x \in \mathbb{R}.$

Assumption A.2.1 insures that for any integer $q \ge 1$, there exists $C_q > 0$ such that

$$\mathbb{E}_X \left[ \sup_{t \in [0,1]} |X_t|^q \right] \le C_q.$$

We also assume that the following Novikov's criterion is fulfilled [86, Prop. (1.15) p. 308].

**Assumption A.2.2. (Novikov's condition)** For all $i \in \mathcal{Y}$, we have

$$\mathbb{E}_X \left[ \exp \left( \frac{1}{2} \int_0^1 \frac{b_i^{*2}}{\sigma^{*2}} (X_s) ds \right) \right] < +\infty.$$

In particular, this assumption allows to apply Girsanov's theorem that is a key ingredient to derive a characterization of the Bayes classifier in the next section.

## A.2.2 Bayes Classifier

The Bayes classifier $g^*$ is a minimizer of the misclassification risk over $\mathcal{G}$

$$g^* \in \arg \min_{g \in \mathcal{G}} \mathcal{R}(g),$$

and is expressed as

$$g^*(X) \in \underset{i \in \mathcal{Y}}{\operatorname{argmax}} \pi_i^*(X), \quad \text{with } \pi_i^*(X) := \mathbb{P}_{(X,Y)}(Y = i | X).$$

The following result of [29] provides a closed form of the conditional probabilities $\pi_i^*, i \in \mathcal{Y}$.

**Proposition A.2.3.** [29] Under Assumptions A.2.1, A.2.2, for all $i \in \mathcal{Y}$, we define

$$F_i^*(X) := \int_0^1 \frac{b_i^*}{\sigma^{*2}}(X_s)dX_s - \frac{1}{2} \int_0^1 \frac{b_i^{*2}}{\sigma^{*2}}(X_s)ds.$$

Under Assumptions A.2.1, A.2.2, for each $i \in \mathcal{Y}$, the conditional probability $\pi_i^*$ is given as follows:

$$\pi_i^*(X) = \phi_i^*(\mathbf{F}^*(X)),$$

where $\mathbf{F}^* = (F_1^*, \dots, F_k^*)$, and $\phi_i^* : (x_1, \dots, x_K) \mapsto \frac{\mathbf{p}_i^* e^{x_i}}{\sum_{k=1}^K \mathbf{p}_k^* e^{x_k}}$ are the softmax functions.

The above proposition provides an explicit dependency of the Bayes classifier on the unknown parameters $\mathbf{b}^*, \sigma^*, \text{ and } \mathbf{p}^*$. Hence, it naturally suggests to build plug-in type estimators $\hat{g}$ of the Bayes classifier $g^*$, relying on estimators of the unknown parameters. In this way, we aim at building an empirical classifier whose misclassification risk is closed to the minimum risk which is reached by the Bayes classifier. The following section is devoted to the presentation of the classification procedure.

## A.3 Classification procedure: a plug-in approach

Let $n \ge 1$ be an integer, and $\Delta_n = 1/n$ the time step which defines the regular grid of the observation time interval $[0, 1]$. Let us assume now that an observation is a couple $(\bar{X}, Y)$, with $\bar{X} := (X_{k\Delta_n})_{0 \le k \le n}$ a high frequency sample path coming from $(X_t)_{t \in [0,1]}$ a solution of Equation (A.1), and $Y$ its associated label. We also introduce, for $N \ge 1$, a learning dataset $\mathcal{D}_N = \{(\bar{X}^j, Y_j), j \in \{1, \dots, N\}\}$ which consists of $N$ independent copies of $(\bar{X}, Y)$. The asymptotic framework is such that $N$ and $n$ tend to infinity.

Based on $\mathcal{D}_N$ we build a classification procedure that relies on the result of Proposition A.2.3. Our classifier uses the knowledge of the class $Y_j$ for the path $X^j$, placing our work in the frame of supervised learning. The procedure is formally described in Section A.3.1 and Section A.3.2 while its statistical properties are provided in Section A.3.3.
### A.3.1 Classifier and excess risk

As suggested by Proposition A.2.3, based on $\mathcal{D}_N$, we first build estimators $\hat{b} = (\hat{b}_1, \dots, \hat{b}_K)$, and $\hat{\sigma}^2$ of $b^*$ and $\sigma^{*2}$ respectively. Besides, we consider the empirical estimators of $\mathfrak{p}_i^*, i = 1, \dots, K$:

$$ \hat{\mathfrak{p}}_i = \frac{1}{N} \sum_{j=1}^{N} \mathbb{1}_{\{Y_j=i\}}. \qquad (A.2) $$

Then, in a second step, we introduce the discretized estimator of $\mathbf{F}^*$

$$ \hat{\mathbf{F}} = (\hat{F}_1, \dots, \hat{F}_K), \quad \text{with} \quad \hat{F}_i(X) = \sum_{k=0}^{n-1} \left( \frac{\hat{b}_i}{\hat{\sigma}^2} (X_{k\Delta}) (X_{(k+1)\Delta} - X_{k\Delta}) - \frac{\Delta}{2} \frac{\hat{b}_i^2}{\hat{\sigma}^2} (X_{k\Delta}) \right). \qquad (A.3) $$

Finally, considering the functions $\hat{\phi}_i : (x_1, \dots, x_K) \mapsto \frac{\hat{\mathfrak{p}}_i e^{x_i}}{\sum_{k=1}^K \hat{\mathfrak{p}}_k e^{x_k}}$, we naturally define the resulting plug-in classifier $\hat{g}$ as

$$ \hat{g}(X) \in \underset{i \in \mathcal{Y}}{\operatorname{argmax}} \hat{\pi}_i(X), \quad \text{with} \quad \hat{\pi}_i(X) = \hat{\phi}_i(\hat{\mathbf{F}}(X)). \qquad (A.4) $$

Hereafter, we establish that the consistency of the plug-in classifier $\hat{g}$ can be obtained through an empirical distance between estimators $\hat{b}$, and $\hat{\sigma}^2$ and the true functions $b^*$, and $\sigma^{*2}$ respectively. This distance relies on the empirical norm defined for $h: \mathbb{R} \to \mathbb{R}$ as

$$ \|h\|_{n,i}^2 := \mathbb{E}_{X|Y=i} \left[ \frac{1}{n} \sum_{k=0}^{n-1} h^2(X_{k\Delta}) \right]. $$

We also introduce the general empirical norm $\|\cdot\|_n$ which, for any function $h$, is

$$ \|h\|_n^2 := \mathbb{E}_X \left[ \frac{1}{n} \sum_{k=0}^{n-1} h^2(X_{k\Delta}) \right]. $$

Let us begin with a proposition which establishes a closed formula of the excess risk in multiclass classification.

**Proposition A.3.1 ([29]).** Let $g$ be a classifier. The following holds

$$ \mathcal{R}(g) - \mathcal{R}(g^*) = \mathbb{E} \left[ \sum_{i=1}^{K} \sum_{j \neq i} |\pi_i^*(X) - \pi_j^*(X)| \mathbb{1}_{\{g(X)=j, g^*(X)=i\}} \right]. $$

The proof of this result is omitted and can be found for instance in [29]. From the result of Proposition A.3.1, and upper-bounding the indicator function by 1, we take advantage of the Lipschitz property of the softmax functions $(\phi_i^*)_{i=1,\dots,K}$ that define the probabilities $(\pi^*(X))_{i=1,\dots,K}$ to majorate the excess risk of an empirical classifier $\hat{g}$ based on $\hat{b} = (\hat{b}_1, \dots, \hat{b}_K)$ and $\hat{\sigma}^2$ by the respective risks of estimation of estimators $\hat{b}_i$ and $\hat{\sigma}^2$. Let us now announce the main result on the excess risk of a plug-in type classifier.

**Theorem A.3.2.** Assume $N$ and $n$ fixed (and large). Grant Assumptions A.2.1, A.2.2. Assume that there exists $b_{\max}$, $\sigma_0^2 > 0$ such that for all $x \in \mathbb{R}$

$$ \max_{i \in \mathcal{Y}} |\hat{b}_i(x)| \le b_{\max} \quad \text{and} \quad \hat{\sigma}^2(x) \ge \sigma_0^2. \qquad (A.5) $$

Then the classifier $\hat{g}$ defined in Equation (A.4) satisfies

$$ \mathbb{E}[\mathcal{R}(\hat{g}) - \mathcal{R}(g^*)] \le C \left( \sqrt{\Delta_n} + \frac{1}{\mathfrak{p}_0 \sqrt{N}} + \mathbb{E} \left[ b_{\max} \sigma_0^{-2} \sum_{i=1}^{K} \|\hat{b}_i - b_i^*\|_n \right] + \mathbb{E} \left[ \sigma_0^{-2} \|\hat{\sigma}^2 - \sigma^{*2}\|_n \right] \right), $$

where $C > 0$ is a constant which depends on $b^*$, $\sigma^*$, and $K$.
Theorem A.3.2 highlights that the excess risk of the plug-in classifier depends on the discretization error which is of order $\Delta_n^{-1/2}$, the $L_2$ error of $\hat{\rho}$ which is of order $N^{-1/2}$, and the estimation error of $\hat{b}$ and $\hat{\sigma}^2$ assessed through the empirical norm $\|\cdot\|_n$. Therefore, a straightforward consequence of Theorem A.3.2 is that consistent estimators of $b^*$, and $\sigma^{*2}$ yield the consistency of plug-in classifier $\hat{g}$. Notice that the additional assumption (A.5) does not require that the true functions $b_i^*$'s are bounded, only their estimators should be. For the difference between $b_i^*$ and $\hat{b}_i$ to remain controlled in the norm $\|\cdot\|_n$, it is necessary that the process $X$ rests with high probability in a compact region of $\mathbb{R}$. The next section is devoted to the construction of consistent estimators of both drift and diffusion coefficients.

## A.3.2 Estimators of drift and diffusion coefficients

In this section, we provide consistent estimators $\hat{b}$, and $\hat{\sigma}^2$, implying the consistency of the associated plug-in classifier. These estimators are defined as minimum contrast estimators under an $l_2$-constraint on a finite dimensional vector space spanned by the $B$-spline basis, but other families of nonparametric estimators could have been chosen as well. In particular, to ensure statistical guarantees on $\mathbb{R}$, the considered estimators are built on a large intervals parameterized by the number $N$ of sample paths, and that tends to the whole real line as $N$ goes to infinity.

### Spaces of approximation

Let $A, \Xi > 0$, and $M \ge 1$. Let $u = (u_{-M}, \dots, u_{\Xi+M})$, a sequence of knots of the compact interval $[-A, A]$ such that

$$u_{-M} = \dots = u_{-1} = u_0 = -A, \quad \text{and} \quad u_{\Xi} = u_{\Xi+1} = \dots = u_{\Xi+M} = A. \\ \forall \ell \in [0, \Xi], \quad u_{\ell} = -A + \frac{2\ell A}{\Xi}.$$

Let us consider the $B$-spline basis $(B_{-M}, \dots, B_{\Xi+M})$ of order $M$ defined by the knots sequence $u$. For the construction of the $B$-spline and its properties, we refer for instance to [57]. Let us mention that the considered $B$-spline functions are nonnegative and $M-1$ continuously differentiable on $[-A, A]$ and are zero outside $[-A, A]$. Besides, for all $x \in [-A, A]$, we have that $\sum_{\ell=-M}^{\Xi-1} B_\ell(x) = 1$. Now, we introduce the space of approximation $S_{\Xi,M}$ defined as

$$S_{\Xi,M} := \left\{ \sum_{\ell=-M}^{\Xi-1} a_\ell B_\ell, \quad \|a\|_2^2 \le (\Xi + M)A^2 \log(N) \right\}, \quad (A.6)$$

where $\|a\|_2^2 = \sum_{\ell=-M}^{\Xi-1} a_\ell^2$ is the usual $l_2$-norm. Note that $A$ can depend on the size $N$ of the learning sample and tend to infinity as $N \to \infty$. The introduction of the constraint space $S_{\Xi,M}$ is motivated by two facts. The first one is the following important property of spline approximations, inspired by the related properties for the Hölder functions (see [57]):

**Proposition A.3.3.** Let $h$ be a $L$-lipschitz function. Then there exists $\tilde{h} \in S_{\Xi,M}$, such that

$$|\tilde{h}(x) - h(x)| \le C \frac{A}{\Xi}, \quad \forall x \in [-A, A],$$

where $C > 0$ depends on $L$, and $M$.

The second one is that the set of functions $S_{\Xi,M}$ is a totally bounded class, in the following sense [32, Chapter 28]. According to [30], for each $\varepsilon \in (0, 1)$ and for $N$ large enough, there exists an $\varepsilon$-net $\tilde{S}_\varepsilon$ of $S_{\Xi,M}$ w.r.t. to the supremum norm $\|\cdot\|_\infty$ such that

$$\log(\mathrm{card}(\tilde{S}_\varepsilon)) \le C_M \Xi \log\left(\frac{\Xi}{\varepsilon}\right).$$

It shows that the complexity of $S_{\Xi,M}$ given in Equation (A.6) is parametric which is particularly appealing in order to apply concentration inequalities.
## Minimum contrast estimators

In this section, we propose two estimators of $b^*$, and $\sigma^{*2}$ which lead to a plug-in classifier that exhibits appealing properties. The construction of the estimators $\hat{b}$, and $\hat{\sigma}^2$ relies on the minimization of a least squares contrast function over the space $S_{\Xi,M}$. They are both based on the observed increments of the process $X$.

**Estimator of the drift functions.** Let $i \in \mathcal{Y}$ and $N_i := \sum_{j=1}^N 1_{\{Y_j=i\}}$ a random variable of Binomial distribution with parameters $(N, \mathfrak{p}_i^*)$. We define the random set $\mathcal{I}_i := \{j, Y_j = i\} = \{i_1, \dots, i_{N_i}\}$ and consider the dataset $\{\tilde{X}^j, j \in \mathcal{I}_i\}$ of size $N_i$ composed of the observations of the class $i$. Hereafter, we work conditional on $(1_{\{Y_1=i\}}, \dots, 1_{\{Y_{N_i}=i\}})$, on the event $\{N_i > 1\}$. Hence, $N_i$ is viewed as a deterministic variable such that $N_i > 1$. In this context, we set for all $i \in \mathcal{Y}$, $A = A_{N_i} > 0$ and $\Xi = K_{N_i} > 0$ where $(A_{N_i})$ and $(K_{N_i})$ are increasing sequences of $N_i$. The first estimator $\tilde{b}_i$ of $b_i^*$ is defined as

$$ \tilde{b}_i \in \arg \min_{h \in S_{K_{N_i}, M}} \frac{1}{n N_i} \sum_{j \in \mathcal{I}_i} \sum_{k=0}^{n-1} (Z_{k \Delta_n}^j - h(X_{k \Delta_n}^j))^2 1_{\{N_i > 0\}}, \quad \text{with} \quad Z_{k \Delta_n}^j := \frac{(X_{(k+1)\Delta_n}^j - X_{k\Delta_n}^j)}{\Delta_n}. \quad (A.7) $$

Then, to fit the assumption of Theorem A.3.2, rather than $\tilde{b}_i$, we consider its thresholded counterpart

$$ \hat{b}_i(x) := \tilde{b}_i(x) 1_{\{|\tilde{b}_i(x)| \le A_{N_i} \log^{1/2}(N)\}} + \operatorname{sgn}(\tilde{b}_i(x)) A_{N_i} \log^{1/2}(N) 1_{\{|\tilde{b}_i(x)| > A_{N_i} \log^{1/2}(N)\}}. \quad (A.8) $$

Note that the value of the threshold $A_{N_i} \log^{1/2}(N)$ corresponds to the bound $b_{\max}$ in (A.5). Although this bound depends on $N$, Theorem A.3.2 can be applied, but to ensure the consistency of the classifier, we now have to prove that the estimation rate for $\hat{b}_i$ decreases sufficiently fast.

**Estimator of the diffusion coefficient.** The construction of the estimator of $\sigma^{*2}$ follows the same lines. However, since the diffusion coefficient is the same for all classes, we can use the whole dataset $\mathcal{D}_N$ to build its estimator with $A = \tilde{A}_N$, $\Xi = \tilde{K}_N$ and $(\tilde{A}_N)$, $(\tilde{K}_N)$ are increasing sequences of $N$. More precisely, we define

$$ \tilde{\sigma}^2 \in \argmin h \in S_{\tilde{K}_N, M} \frac{1}{nN} \sum_{j=1}^{N} \sum_{k=0}^{n-1} (U_{k\Delta_n}^j - h(X_{k\Delta_n}^j))^2, \quad \text{with} \quad U_{k\Delta_n}^j = \frac{(X_{(k+1)\Delta_n}^j - X_{k\Delta_n}^j)^2}{\Delta_n}. \quad (A.9) $$

Finally, as for the drift estimator we consider the truncated version $\hat{\sigma}^2$ as

$$ \hat{\sigma}^2(x) := \tilde{\sigma}^2(x) 1_{\{a_N \le \tilde{\sigma}^2(x) \le \tilde{A}_N \log^{1/2}(N)\}} + \tilde{A}_N \log^{1/2}(N) 1_{\{\tilde{\sigma}^2(x) > \tilde{A}_N \log^{1/2}(N)\}} + a_N 1_{\{\tilde{\sigma}^2(x) \le a_N\}}, \quad (A.10) $$

with $a_N = 1/\log(N)$. Although this constraint does not appear in Theorem A.3.2, it remains natural in view of Assumption A.2.1 (ii). We will impose that $\hat{\sigma}^2$ is bounded by $\tilde{A}_N \log^{1/2}(N)$ to derive its consistency.

### A.3.3 A general consistency result

In this section, we establish the consistency of the empirical classifier based on the estimators presented in the previous section. We first provide rates of convergence for the estimators of both the drift and the diffusion coefficients.

**Theorem A.3.4.** Let $i \in \mathcal{Y}$, and set $A_{N_i} = \log^2(N_i)$ conditional on the event $\{N_i > 1\}$, and $\tilde{A}_N = \log^2(N)$. Assume that Assumptions A.2.1, A.2.2 are satisfied. Considering the estimator $\hat{b}_i$ of $b_i^*$ (A.8) and the estimator $\hat{\sigma}^2$ of $\sigma^{*2}$ (A.10), set $K_{N_i} \propto (N_i \log(N_i))^{1/5}$ for $\hat{b}_i$, and $\tilde{K}_N \propto (N \log(N))^{1/5}$ for $\hat{\sigma}^2$. For $N$, $N_i$ then n large enough, such that $\Delta_n = O(1/N)$, we have

$$ \mathbb{E}[\|\hat{b}_i - b_i^*\|_{n,i}] \le C_1 \left(\frac{\log^4(N)}{N}\right)^{1/5}, \quad \text{and} \quad \mathbb{E}[\|\hat{\sigma}^2 - \sigma^{*2}\|_n] \le C_2 \left(\frac{\log^4(N)}{N}\right)^{1/5}, $$

where $C_1, C_2 > 0$ are constants which depend on $L_0$, $\mathfrak{p}_0$, and $K$.
Considering the estimation of the drift functions $b_i^*$, when $n, N_i \to \infty$ a.s., we control the risk

$$ \mathbb{E} [\|\hat{b}_i - b_i^*\|_{n,N_i}^2] = \mathbb{E} \left[ \frac{1}{nN} \sum_{j=j_1}^{j_{N_i}} \sum_{k=0}^{n-1} (\hat{b}_i - b_i^*)^2 (X_{k\Delta_n}^{(j)}) \right], $$

using Equation (A.7), the properties of the constrained subspace $S_{K_{N_i}, M}$ together with that of functions of the spline basis (see proof of Theorem A.3.4 for more details). Then, the control of the risk

$$ \mathbb{E} [\|\hat{b}_i - b_i^*\|_{n,i}^2] := \mathbb{E} \left[ \sum_{k=0}^{n-1} (\hat{b}_i - b_i^*)^2 (X_{k\Delta_n}) \right], $$

is obtained from the previous risk $\mathbb{E} [\|\hat{b}_i - b_i^*\|_{n,N_i}^2]$ using, for all function $h$ such that $\|h\|_{\infty} = O(L_N)$ with $(L_N)$ an increasing sequence of $N$, the following result

$$ \mathbb{E} [\|h\|_n^2] - 2\mathbb{E} [\|h\|_{n,N}^2] = O\left(\frac{K^*L_N \log(N)}{N}\right), $$

proved in [30], Lemma A.2.

Several comments can be made. First, we obtain a general rate of convergence for the estimation on $\mathbb{R}$ for both drift and diffusion coefficient functions under mild assumptions. This rate is, up to a logarithmic factor, of order $N^{-1/5}$. Hence, it extends the result of Theorem 3.3 in [30], where only consistency of drift estimators is obtained. In particular, a difficulty in establishing the convergence rate on $\mathbb{R}$ is to control the exit probabilities from the intervals $(-A_{N_i}, A_{N_i})$ and $(-\tilde{A}_N, \tilde{A}_N)$, which are provided here by careful estimates for the transition densities following [55].

This result together with Theorem A.3.2 yields the consistency of the plug-in classifier

$$ \hat{g} := \hat{g}_{\hat{\mathbb{P}}, \hat{\mathbb{B}}, \hat{\sigma}^2}, \qquad (A.11) $$

where the unknown parameters are replaced by their estimators in Equation (A.4). However, application of Theorem A.3.2 requires the consistency of the estimator $\hat{b}_i$ in terms of empirical norm $\|\cdot\|_n$ and not in terms of norm $\|\cdot\|_{n,i}$. To circumvent this issue, we can use a change of probability to get rid of the conditioning on $Y = i$. For this purpose, we take advantage of Lemma A.7.3 and A.7.4 to derive precise control of the transition density of the process $X$ conditioned on $Y = i$, and then to establish the consistency of the plug-in classifier.

**Theorem A.3.5.** Grant Assumptions A.2.1, A.2.2. For $N$ large enough, set $\Delta_n = O(1/N)$, $\tilde{A}_N = \log(N)$ and $\tilde{K}_N = (N \log(N))^{1/5}$. Moreover, for each $i \in \mathcal{Y}$, on the event $\{N_i > 1\}$, $A_{N_i} = \log(N_i)$ and $K_{N_i} \propto (N_i \log(N_i))^{1/5}$. Then, the classifier $\hat{g}$ satisfies

$$ \mathbb{E} [\mathcal{R}(\hat{g}) - \mathcal{R}(g^*)] \xrightarrow{N \to \infty} 0. $$

The consistency of our classification procedure is obtained under very mild assumptions. The study of the rates of convergence requires more structural assumptions. In the following section, we obtain rates of convergence of the plug-in classifier under different kind of assumptions.

### A.3.4 General rate of convergence for bounded drift function

In this section, we study the general rate of convergence of the proposed method described in Section A.3.1 under the additional assumption that the drift functions of the considered mixture model are bounded, no additional assumptions being made on the diffusion coefficient.

Let us consider the following assumption.

**Assumption A.3.6.** There exists $C_{b^*}$ such that

$$ \max_{i \in \mathcal{Y}} \|b_i^*\|_{\infty} \leq C_{b^*}. $$
Let $i, j \in \mathcal{Y}^2$ with $i \neq j$. The following property allows to upper bound the expectation conditional on $\{Y = i\}$ by the expectation conditional on $\{Y = j\}$. This happens to be the cornerstone to derive rates of convergence for our procedure.

**Proposition A.3.7.** Under Assumptions A.2.1, A.2.2, and A.3.6, we have for all $i, j \in \mathcal{Y}^2$ such that $i \neq j$, and $N$ large enough

$$ \|\hat{b}_i - b_i^*\|_{n,j}^2 \le C \exp\left(\sqrt{c \log(N)}\right) \|\hat{b}_i - b_i^*\|_{n,i}^2 + C \frac{A^2 \log(N)}{N}, $$

where $C, c > 0$ depend on $C_{b^*}, \sigma_1$, and $\sigma_0$.

A crucial consequence of this result is that in particular the empirical norms $\|\cdot\|_{n,i}$, $i \in \mathcal{Y}$, are now equivalent up to a factor of order $\exp(\sqrt{c \log(N)})$. Notice that for all $r_1, r_2 > 0$,

$$ \log^{r_1}(N) = o\left(\exp\left(\sqrt{c \log(N)}\right)\right), \quad \text{and} \quad \exp\left(\sqrt{c \log(N)}\right) = o(N^{r_2}). \quad (A.12) $$

In particular, the factor $\exp(\sqrt{c \log(N)})$ is negligible with respect to any power of $N$. Therefore, combining Theorem A.3.2, A.3.4, and Proposition A.3.7, we are able to give the rate of convergence for our procedure (when the drift coefficients are globally Lipschitz and bounded).

**Theorem A.3.8.** Grant Assumptions A.2.1, A.2.2, and A.3.6. Set $\tilde{A}_N = \log(N)$ and $\tilde{K}_N \propto (N \log(N))^{1/5}$. Moreover, for each class $i \in \mathcal{Y}$, on the event $\{N_i > 1\}$, $A_{N_i} = \log(N_i)$ and $K_{N_i} \propto (N_i \log(N_i))^{1/5}$. The plug-in classifier $\hat{g}$ given in Equation (A.11), provided that $\Delta_n = O(N^{-1})$ and $N$ large enough, satisfies

$$ \mathbb{E}[\mathcal{R}(\hat{g}) - \mathcal{R}(g^*)] \le C \exp\left(\sqrt{c \log(N)}\right) N^{-1/5}, $$

where $C > 0$ depends on $C_{b^*}, \sigma_1$, and $\sigma_0$.

Leveraging the result of Theorem A.3.4 and Proposition A.3.7, we obtain a rate of convergence which is of order $N^{-1/5}$ up to the extra factor $\exp(\sqrt{c \log(N)})$. Note that the optimal rate of convergence obtained when the estimation of drift function is done over on a compact set is of order $N^{-1/3}$ w.r.t. $\|\cdot\|_n$ rather than $N^{-1/5}$ (see [30]). Here, this slower rate is mainly due to the fact that our procedure requires a control of the drift estimators over $\mathbb{R}$.

In the next section, we show that when $\sigma^*$ is constant and assumed to be known, we derive faster rates of convergence. In particular, under Assumption A.3.6, we show that our plug-in procedure achieves a rate of convergence of order $N^{-1/3}$. Lastly, note that Theorem A.3.8 can be easily extended to higher order of regularity for the drift functions (e.g. Hölder with regularity $\beta > 1$). In this case, the obtained rate of convergence is of order $N^{-\beta/(2\beta+3)}$.

## A.4 Classifier's rate of convergence with known diffusion coefficient

In this section, we consider that the diffusion coefficient is known and constant, and we derive faster rates of convergence of the classification procedure. For sake of simplicity, we choose $\sigma^* = 1$. In this case, our plug-in procedure only involves the estimation of the drift function $\hat{b}$. Hence, the plug-in classifier now writes as $\hat{g} = \hat{g}_{p, \hat{b}, 1}$.

In order to derive a general rate of convergence as a function of the drift regularity, we consider the following smoothness assumption [95], which is a subset of Lipschitz functions.

**Assumption A.4.1.** For all $i \in \mathcal{Y}$, $b_i^*$ is Hölder with regularity parameter $\beta \ge 1$.
### A.4.1 Rates of convergence for drift estimators

Let $i \in \mathcal{Y}$. The study of the rates of convergence of the estimator $\hat{b}_i$ relies on the properties of the matrix $\Psi_{K_{N_i}} \in \mathbb{R}^{(K_{N_i}+M)^2}$ defined by

$$ \Psi_{K_{N_i}} := \left( \frac{1}{n} \sum_{k=0}^{n-1} \mathbb{E}_{X|Y=i} [B_{\ell}(X_{k\Delta}^i) B_{\ell'}(X_{k\Delta}^i)] \right)_{\ell, \ell' \in [-M, K_{N_i}-1]} . \quad (A.13) $$

Note that for $t \in S_{K_{N_i}, M}$, $t = \sum_{i=-M}^{K_{N_i}-1} a_i B_{i,M,u}$, we have the relation

$$ \|t\|_{n,i}^2 = a' \Psi_{K_{N_i}} a, \text{ with } a = (a_{-M}, \dots, a_{K_{N_i}-1})'. $$

Let us remind the reader that for a matrix $P$, the operator norm $\|P\|_{op}$ is defined as the square root of the largest eigenvalue of the matrix $P'P$. Besides, if $P$ is symmetric, its norm is equal to its largest eigenvalue. The matrix $\Psi_{K_{N_i}}$ satisfies the following property.

**Lemma A.4.2.** Conditional on $(1_{\{Y_1=i\}}, \dots, 1_{\{Y_N=i\}})$, on the event $\{N_i > 1\}$, the matrix $\Psi_{K_{N_i}}$ given in Equation (A.13) satisfies

(i) if $K_{N_i} \ge 1$, $\Psi_{K_{N_i}}$ is invertible,

(ii) under Assumption A.2.1, for $N$ large enough, if $K_{N_i} \le \sqrt{N_i}$, there exists two constants $C, c > 0$ such that

$$ c \frac{K_{N_i}}{A_{N_i}} \exp \left( \frac{A_{N_i}^2}{6} \right) \le \| \Psi_{K_{N_i}}^{-1} \|_{op} \le C \frac{K_{N_i} \log(N_i)}{A_{N_i}} \exp \left( \frac{2}{3} A_{N_i}^2 \right). $$

A major consequence of Lemma A.4.2 is to give the order of $A_{N_i}$ w.r.t. $N_i$ to obtain optimal rates of convergence for the estimation of the drift function $b_i^*$.

For fixed $n$ and $N_i$ in $\mathbb{N}^*$, let us denote

$$ \Omega_{n,N_i,K_{N_i}} := \bigcap_{h \in S_{K_{N_i},M} \setminus \{0\}} \left\{ \left| \frac{\|h\|_{n,N_i}^2}{\|h\|_{n,i}^2} - 1 \right| \le \frac{1}{2} \right\}. $$

As we can see, the empirical norms $\|h\|_{n,N_i}$ and $\|h\|_{n,i}$ of any function $h \in S_{K_{N_i},M} \setminus \{0\}$ are equivalent on the random set $\Omega_{n,N_i,K_{N_i}}$. More precisely, on the set $\Omega_{n,N_i,K_{N_i}}$, for all $h \in S_{K_{N_i},M} \setminus \{0\}$, we have

$$ \frac{1}{2} \|h\|_{n,i}^2 \le \|h\|_{n,N_i}^2 \le \frac{3}{2} \|h\|_{n,i}^2. $$

We use the random set $\Omega_{n,N_i,K_{N_i}}$ to derive a faster rate of the risk $\mathbb{E}[\|\hat{b}_i - b_{A_{N_i},i}^*\|_{n,N_i}^2]$ of the estimator $\hat{b}_i$ of the restriction $b_{A_{N_i},i}^*$ of the drift function $b_i^*$ on the interval $[-A_{N_i}, A_{N_i}]$, with $A_{N_i} > 0$ on the event $\{N_i > 1\}$. We have for $N$ large enough

$$ \begin{aligned} \mathbb{E}[\|\hat{b}_i - b_{A_{N_i},i}^*\|_{n,N_i}^2] &= \mathbb{E}[\|\hat{b}_i - b_{A_{N_i},i}^*\|_{n,N_i}^2 \mathbb{1}_{\Omega_{n,N_i,K_{N_i}}}] + \mathbb{E}[\|\hat{b}_i - b_{A_{N_i},i}^*\|_{n,N_i}^2 \mathbb{1}_{\Omega_{n,N_i,K_{N_i}}^c}] \\ &\le \mathbb{E}[\|\hat{b}_i - b_{A_{N_i},i}^*\|_{n,N_i}^2 \mathbb{1}_{\Omega_{n,N_i,K_{N_i}}}] + 4A_N^2 \log(N) \mathbb{P}(\Omega_{n,N_i,K_{N_i}}^c), \end{aligned} $$

and the probability $\mathbb{P}(\Omega_{n,N_i,K_{N_i}}^c)$ satisfies

$$ \mathbb{P}(\Omega_{n,N_i,K_{N_i}}^c) \le 2(K_{N_i} + M) \exp \left( -C \frac{N_i}{A_{N_i} \|\Psi_{K_{N_i}}^{-1}\|_{op}} \right), \quad (A.14) $$
(see proof of Lemma 2.9.9, Equation (A.77) in Appendix). From Equation (A.14) and Lemma A.4.2, we obtain

$$ \mathbb{P}(\Omega_{n,N_i,K_{N_i}}^c) \le 2(K_{N_i} + M) \exp\left(-C \frac{N_i}{K_{N_i} \log(N_i)} \exp\left(-\frac{2}{3} A_{N_i}^2\right)\right). \quad (A.15) $$

Notably, conditional on $(1_{\{Y_1=i\}}, \dots, 1_{\{Y_N=i\}})$ and on the event $\{N_i > 1\}$, if $K_{N_i}$ is of order $N_i^{1/(2\beta+1)}$ (up to some extra logarithmic factors), and $A_{N_i}$ is chosen such that the upper-bound of $\mathbb{P}(\Omega_{n,N_i,K_{N_i}}^c)$ is dominated by $K_{N_i}/N_i$ as $N$ tends to infinity, then the drift estimator converges as $N_i^{-2\beta/(2\beta+1)}$ w.r.t. $\|\cdot\|_{n,i}^2$. Interestingly, this is the same rate of convergence obtained in [30] when the estimation of the drift function is performed over a fixed compact interval. From this remark, if $K_{N_i}$ is of order $\log^{-5/2}(N_i)N_i^{1/(2\beta+1)}$, and $A_{N_i} \le \sqrt{\frac{3\beta}{2\beta+1}\log(N_i)}$, we deduce from Equation (A.15) that there exists a constant $C > 0$ such that

$$ \mathbb{P}(\Omega_{n,N_i,K_{N_i}}^c) \le 2(K_{N_i} + M) \exp(-C \log^{3/2}(N_i)), $$

and the desired result is obtained since $N_i \to \infty$ a.s. as $N \to \infty$. Furthermore, the lemma shows that the order of $A_{N_i}$ is tight. Indeed, for another choice of $A_{N_i}$ such that

$$ \frac{A_{N_i}}{\sqrt{\log(N_i)}} \to +\infty \quad \text{as} \quad N \to +\infty, $$

then, from Equation (A.15), the upper-bound of $\mathbb{P}(\Omega_{n,N_i,K_{N_i}}^c)$ is of order $K_{N_i}$ since

$$ \exp\left(-C \frac{N_i}{K_{N_i} \log(N_i)} \exp\left(-\frac{2}{3} A_{N_i}^2\right)\right) \to 1 \quad \text{a.s. as } N \to \infty, $$

and the convergence of $\mathbb{P}(\Omega_{n,N_i,K_{N_i}}^c)$ toward 0 is no longer guaranteed. Based on this observation, the next result establishes the rates of convergence for our proposed drift estimator on the event $\{N_i > 1\}$.

**Theorem A.4.3.** Let Assumptions A.2.1, A.2.2 and A.4.1 be satisfied. Let $b_{A_{N_i},i}^* = b_i^*1_{[-A_{N_i},A_{N_i}]}$ defined on the event $\{N_i > 1\}$. If $A_{N_i} \le \sqrt{\frac{3\beta}{2\beta+1}\log(N_i)}$, $K_{N_i} \propto (\log^{-5/2}(N_i)N_i^{1/(2\beta+1)})$, and $\Delta_n = O(N^{-1})$. Then for all $i \in \mathcal{Y}$

$$ \mathbb{E}\left[\left\|\hat{b}_i - b_{A_{N_i},i}^*\right\|_{n,i}^2 1_{N_i>1}\right] \le C \log^{6\beta}(N) N^{-2\beta/(2\beta+1)}, $$

where C is a constant which depends on $b^*$.

The above result shows that for a proper choice of $A_{N_i}$ the drift estimators $\hat{b}_i$ achieves, up to a logarithmic factor, the minimax rates of convergence w.r.t. $\|\cdot\|_{n,i}$ (see Theorem 4.7 in [30]). Notably, Theorem A.4.3 extends results obtained in [30] to the estimation of the drift function on an interval which depends on $N$.

In Section A.4.2 and Section A.4.3, we exploit this result to derive rates of convergence for the plug-in classifier $\hat{g}$ defined as follows. On the event $\{\min_{i \in \mathcal{Y}} N_i > 1\}$, we consider the estimators $\hat{b}$ presented in Section A.3.2, and define the plug-in classifier $\hat{g} = \hat{g}_{\hat{p},\hat{b},1}$. On the complementary event $\{\min_{i \in \mathcal{Y}} N_i \le 1\}$, we simply set $\hat{g} = 1$.

## A.4.2 Rates of convergence: bounded drift functions

In this section, we assume that additionally to $\sigma^* = 1$, Assumption A.3.6 is fulfilled (the drift function is bounded). Hence, we can use Proposition A.3.7, and apply Theorem A.4.3 to derive rates of convergence for plug-in estimator $\hat{g}$.
**Theorem A.4.4.** Grant Assumptions A.2.1, A.2.2, A.3.6, A.4.1. Assume that for all $i \in \mathcal{Y}$, on the event $\{N_i > 1\}$, $A_{N_i} = \sqrt{\frac{6\beta}{2\beta+1}\log(N_i)}$ and $K_{N_i} \propto (\log^{-5/2}(N_i)N_i^{1/(2\beta+1)})$, and $\Delta_n = O(N^{-1})$. Then the plug-in classifier $\hat{g} = \hat{g}_{\hat{\mathbf{b}}_{1,1}}$ satisfies

$$ \mathbb{E}[\mathcal{R}(\hat{g}) - \mathcal{R}(g^*)] \le C \exp\left(\sqrt{c \log(N)}\right) N^{-\beta/(2\beta+1)}, $$

where $C, c > 0$ are constants depending on $\mathbf{b}^*, \beta, K$ and $\mathfrak{p}_0$.

The above theorem shows that the plug-in classifier $\hat{g}$ achieves faster rates of convergence than in the case where $\sigma^*$ is unknown (see Theorem A.3.8). Notably, the obtained rate is of the same order, up to a factor of order $\exp(\sqrt{c \log(N)})$, than the rates of convergence provided in [47] in the framework of binary classification of functional data where the observation are assumed to come from a white noise model. In their setting, $\sigma^* = 1$ and the drift functions depend only on the observation time interval, which is also assumed to be $[0, 1]$. Therefore, our specific setup is more challenging since the drift functions are space-dependent which involves to deal with estimation of function on a non-compact interval. Finally, it is worth noting that, up to the $\exp(\sqrt{c \log(N)})$ factor, the rate of convergence provided in Theorem A.4.4 is the same as the minimax rates in the classical classification framework where the feature vector $X$ belongs to $\mathbb{R}^d$ and that $X$ admits a lower bounded density [101, 4].

### A.4.3 Rates of convergence: when the drift functions are re-entrant

In this section, we study performance of the plug-in classifier when the drift functions are not necessarily bounded. In this context, rates of convergence are obtained under the following assumption.

**Assumption A.4.5.** (re-entrant drift function) For each label $i \in \mathcal{Y}$, there exists $c_0 > 4$ and $K_0 \in \mathbb{R}$ such that

$$ \forall x \in \mathbb{R}, \quad b_i^*(x)x \le -c_0x^2 + K_0. $$

An important consequence of this assumption is that there exists $C > 0$ (see Proposition 1.1 in [55]) such that

$$ \mathbb{E}[\exp(4|X_t|^2)] \le C, \qquad (A.16) $$

which yields a better bound on the tail probability $\mathbb{P}(|X_t| \ge A)$ for $A > 0$. It worth noting that under Assumption A.4.5, the drift functions are not bounded. Hence, we can not take advantage of Proposition A.3.7 to derive rates of convergence. Nonetheless, we obtain the following result.

**Theorem A.4.6.** Grant Assumptions A.2.1, A.2.2, A.4.1, A.4.5. Assume that for all $i \in \mathcal{Y}$, on the event $\{N_i > 1\}$, $A_{N_i} = \sqrt{\frac{3\beta}{2\beta+1}\log(N_i)}$ and $K_{N_i} \propto (\log^{-5/2}(N_i)N_i^{1/(2\beta+1)})$, and $\Delta_n = O(N^{-1})$. Then, the plug-in classifier $\hat{g}$ satisfies

$$ \mathbb{E}[\mathcal{R}(\hat{g}) - \mathcal{R}(g^*)] \le C \log^{3\beta+1}(N) N^{-3\beta/4(2\beta+1)}. $$

The above theorem shows that the rate of convergence of the plug-in classifier is, up to a logarithmic factor, of order $N^{-3\beta/4(2\beta+1)}$. Therefore, this rate of convergence is slightly slower than the one provided in Theorem A.4.4. It is mainly due to the fact that under Assumption A.4.5, Proposition A.3.7 does not apply and then, in view of considered assumptions in Theorem A.4.6, we only manage to obtain the following bound,

$$ \forall i, j \in \mathcal{Y}: i \ne j, \quad \mathbb{E}\left[\left\|\hat{b}_i - b_i^*\right\|_{n,j}^2\right] \le CN^{\beta/4(2\beta+1)}\mathbb{E}\left[\left\|\hat{b}_i - b_i^*\right\|_{n,i}^2\right], $$

which is clearly worst than the one obtained in Proposition A.3.7. Interestingly, for $\beta = 1$, we can note that the rates obtained in Theorem A.4.6 are of the same order than the rates of convergence established in [48] in the classification setup where the input vector lies in $\mathbb{R}$ under the assumption that $X$ does not fulfil the strong density assumption (e.g. the density of $X$ is not lower bounded).
## A.5 Simulation study

This section is devoted to numerical experiments that support our theoretical findings. A first part is dedicated to the study of the performance of the plug-in classifier in a setting which meets the assumptions of Section A.3.4. The considered model is presented in Section A.5.1. The implementation of the proposed procedure is discussed in Section A.5.2 while the performances of the plug-in classifier are given in Section A.5.3. Finally, several features of the problem are investigated in Section A.5.4. In particular, we consider the classical Ornstein-Uhlenbeck model, for which assumptions of Section A.3.4 are not fulfilled.

### A.5.1 Models and simulation setting

We fix $K = 3$ classes in the following. Note that, we do not consider larger value of $K$ since the evaluation of the impact of $K$ on the procedure is beyond the scope of this paper. To illustrate the accuracy of the presented plug-in classifier, we investigate the model described in Table A.1. This

<table><caption>Table A.1: *Drift and diffusion coefficients, depending on $\theta \in \Theta = \{1/2, 3/4, (4+\alpha)/4, \alpha \in [1, 12]\}$.*</caption><tr><td>$b_1^*(x)$</td><td>$1/4 + (3/4) \cos^2 x$</td></tr><tr><td>$b_2^*(x)$</td><td>$\theta[1/4 + (3/4) \cos^2 x]$</td></tr><tr><td>$b_3^*(x)$</td><td>$-\theta[1/4 + (3/4) \cos^2 x]$</td></tr><tr><td>$\sigma^*(x)$</td><td>$0.1 + 0.9/\sqrt{1+x^2}$</td></tr></table>

toy model, described in Table A.1, fulfills the assumptions of Section A.3.4. Interestingly, this model allows evaluating the influence of the distance between the drift functions of each of the three classes, on the classification problem, through the parameter $\theta$. Indeed,

$$ \min_{i,j=1,2,3} \|b_i^* - b_j^*\|_{\infty} = \theta, \quad \text{where } \theta \in \Theta = \{1/2, 3/4, (4+\alpha)/4, \alpha \in [1, 12]\}. $$

We investigate the consistency of the empirical classifier using learning samples of size $N \in \{100, 1000\}$ with $n \in \{100, 500\}$ (and thus with $\Delta_n = 1/n$). We use the R-package sde (see [62]) to simulate the solution of the stochastic differential equation corresponding to the chosen model.

Figure A.1 displays simulated trajectories from the proposed model. On the left panel (right panel respectively) the observed learning sample comes from the model with parameter $\theta = 1/2$ ($\theta = 4$ respectively) and each class is represented by one color. We can see from Figure A.1 that the distance between the drift functions strongly impacts the dispersion of the trajectories and leads to a more difficult classification task.

<figure><img src="image_13.png" /><figcaption>Figure A.1: *Dispersion of diffusion paths from model given in Table A.1. Left: $\theta = 1/2$, right: $\theta = 4$ (blue lines $K = 1$, purple lines $K = 2$, green lines $K = 3$); with $N = 100$ and $n = 100$.*</figcaption></figure>
**Performance of the Bayes classifier.** We evaluate the performance of the Bayes classifier $g^*$ with respect to four values of parameter $\theta$ ($\theta \in \{1/2, 3/2, 5/2, 4\}$). To this end, we compute its average error rate over 100 repetitions of the following steps

(i) simulate $D_N$ of size $N = 4000$ with $n = 500$;

(ii) based on $D_N$ compute the misclassification error rate of the discrete counterpart of $g^*$.

Table A.2 provides the mean and standard deviation of the misclassification risk. The obtained results highlight the significant impact of the minimum distance $\theta$, between the drift functions of each class, on the performance of $g^*$. Indeed, as expected, the Bayes classifier is more accurate on our model when parameter $\theta$ is large, especially in the case of separable data ($\theta = 4$). On the contrary, the worst case corresponds to $\theta = 0.5$. In this model, the data are highly ambiguous.

<table><caption>Table A.2: Classification risks of the Bayes classifier $g^*$ w.r.t parameter $\theta$ from learning samples of size $N = 4000$ with $n = 500$.</caption><thead><tr><th></th><th>θ = 1/2</th><th>θ = 3/2</th><th>θ = 5/2</th><th>θ = 4</th></tr></thead><tbody><tr><th>R̂(g*)</th><td>0.49 (0.01)</td><td>0.36 (0.01)</td><td>0.22 (0.01)</td><td>0.11 (0.01)</td></tr></tbody></table>

## A.5.2 Implementation of the plug-in classifier

Hereafter, we briefly describe the implementation of the proposed plug-in classifier. We first estimate the drift functions $b_i^*$, $i = 1, 2, 3$. For each $i \in \{1, 2, 3\}$, the estimator $\hat{b}_i$ is built on the interval $[-A_{N_i}, A_{N_i}]$. Since the drifts (and the diffusion) coefficients are bounded, we can use the construction considered in Section A.3. Therefore, we fix $A_{N_i} = \log(N_i)$, $M = 3$, and divide the learning sample $D_N$ into sub-samples $D_N^i$ of size $N_i$ that contains all diffusion paths belonging to the class $i$. From the sub-sample $D_N^i$, we build estimators $\hat{b}_i$, $i = 1, 2, 3$.

For the construction of the estimator $\hat{b}_i$, we have to choose the dimension parameter $K_{N_i}$. We follow [30], and consider an adaptive choice denoted by $\hat{K}_{N_i}$.

Let us remind the reader that in [30], the adaptive dimension $\hat{K}_{N_i}$ is selected such that $\hat{K}_{N_i}$ is the minimizer of the following penalized contrast

$$ \hat{K}_{N_i} := \underset{K \in \mathcal{K}}{\operatorname{argmin}} \left\{ \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} (\hat{b}_{i,K} - Z_{k\Delta_n}^j)^2 + \operatorname{pen}_b(K) \right\}, \quad (A.17) $$

where $\mathcal{K} \in \{2^q, q \in [0, 5]\}$, and $\hat{b}_{i,K}$ is the drift estimator built on the approximation subspace $S_{K,M}$. Besides, $\operatorname{pen}_b(K) = \kappa(K + M) \log^3(N)/N$ is the penalty function with $\kappa > 0$. We fix the parameter $\kappa = 0.1$ as recommended in [30]. For the estimation of $\sigma^2$, we consider the whole sample $D_N$ and apply the methodology described in Section A.3 with $M = 3$. We follow the same lines to build an adaptive estimator of $\sigma^{2*}$, and choose $\hat{K}_N$ as the minimizer over $\mathcal{K}$ of the following penalized contrast

$$ \hat{K}_N := \underset{K \in \mathcal{K}}{\operatorname{argmin}} \left\{ \frac{1}{Nn} \sum_{j=1}^{N} \sum_{k=0}^{n-1} (\hat{\sigma}_K^2 - U_{k\Delta_n}^j)^2 + \operatorname{pen}_\sigma(K) \right\}, \quad (A.18) $$

where $\hat{\sigma}_K^2$ is the estimator built on $S_{K,M}$, and $\operatorname{pen}_\sigma(K) := \kappa(K + M) \log^3(N)/Nn$ is the penalty function, with $\kappa > 0$. The value of the tuning parameter $\kappa$ is calibrated through an intensive simulation study and chosen equal to $\kappa = 5$.

The function SDEclassif of the R-package SDEclassif, available on github, implements the resulting plug-in classifier.
### A.5.3 Simulation results

The performance of the plug-in classifier $\hat{g}$ is evaluated by repeating 100 times the following steps

1. Simulate learning samples $\mathcal{D}_N$ and $\mathcal{D}_{N'}$ with $N \in \{100, 1000\}$, $N' = 1000$, and $n \in \{100, 500\}$;

2. for each $i \in \{1, 2, 3\}$, from the sub-sample $D_N^i = \{\bar{X}^j, j \in \mathcal{I}_i\}$, select $\hat{K}_N$ minimizing (A.17) and compute the estimator $\hat{b}_{i, \hat{K}_N}$ of $b_i^*$ given in Equation (A.8);

3. from $\mathcal{D}_N$ select $\hat{K}_N$ using Equation (A.18) and compute the estimator $\hat{\sigma}_{\hat{K}_N}^2$ of $\sigma^{*2}$ given in (A.10);

4. based on $\mathcal{D}_N$ compute $\hat{\mathfrak{p}} = (\frac{1}{N} \sum_{j=1}^N 1_{Y_j=1}, \frac{1}{N} \sum_{j=1}^N 1_{Y_j=2}, \frac{1}{N} \sum_{j=1}^N 1_{Y_j=3})$;

5. based on $\mathcal{D}_{N'}$, compute the error rate of the plug-in classifier $\hat{g}$ where $\hat{\mathbf{b}} = (\hat{b}_{1, \hat{K}_N}, \hat{b}_{2, \hat{K}_N}, \hat{b}_{3, \hat{K}_N})$ and $\hat{\sigma}^2 = \hat{\sigma}_{\hat{K}_N}^2$, and $\hat{\mathfrak{p}}$.

From these repetitions, we compute the empirical mean and standard deviation of the error rate of $\hat{g}$. The results are given in Table A.3 and Figure A.2. As expected, from Table A.3 and Table A.2, we can see that the error rate of the plug-in classifier $\hat{g}$ is closed to the error rate of the Bayes classifier. In particular, for $N = 1000$, it performs as well as the Bayes classifier. Note that the length of the paths $n$ does not significantly impact the performance of $\hat{g}$. Moreover, from Figure A.2, we can make similar comments as for the Bayes classifier (see Table A.2), in particular, the accuracy of $\hat{g}$ decreases as parameter $\theta$ increases.

<table><caption>Table A.3: Risks of the plug-in classifier $\hat{g}$ w.r.t. values of parameter $\theta$</caption><thead><tr><th rowspan="2">R̂(ĝ)</th><th colspan="2">n = 100</th><th colspan="2">n = 500</th></tr><tr><th>N = 100</th><th>N = 1000</th><th>N = 100</th><th>N = 1000</th></tr></thead><tbody><tr><td>θ = 1/2</td><td>0.53 (0.05)</td><td>0.50 (0.05)</td><td>0.53 (0.05)</td><td>0.49 (0.05)</td></tr><tr><td>θ = 3/2</td><td>0.39 (0.06)</td><td>0.37 (0.05)</td><td>0.39 (0.05)</td><td>0.36 (0.05)</td></tr><tr><td>θ = 5/2</td><td>0.24 (0.05)</td><td>0.22 (0.04)</td><td>0.25 (0.04)</td><td>0.22 (0.04)</td></tr><tr><td>θ = 4</td><td>0.12 (0.03)</td><td>0.10 (0.03)</td><td>0.11 (0.03)</td><td>0.10 (0.03)</td></tr></tbody></table>

### A.5.4 Ornstein-Uhlenbeck model

In this section, we focus on the influence of the diffusion coefficient $\sigma^*$ on the performance of our plug-in procedure. To this end, we consider the Ornstein-Uhlenbeck diffusion model given in Table A.4 where the diffusion coefficient $\sigma^*$ is constant. Let us notice also that in this model the drift functions are unbounded. We investigate the performance of the plug-in classifier $\hat{g}$ w.r.t. the level

<table><caption>Table A.4: Ornstein-Uhlenbeck mixture model with $K = 3$</caption><tr><td>b<sub>1</sub><sup>*</sup>(x)</td><td>1 - x</td></tr><tr><td>b<sub>2</sub><sup>*</sup>(x)</td><td>-1 - x</td></tr><tr><td>b<sub>3</sub><sup>*</sup>(x)</td><td>-x</td></tr><tr><td>σ<sup>*</sup>(x)</td><td>σ</td></tr></table>

of noise $\sigma^*$. This study is motivated by the fact that, inherently, the diffusion coefficient impacts the dispersion of the trajectories. Therefore, it can lead to separable data when $\sigma^*$ is close to zero, and ambiguous data for large values of $\sigma^*$. Thus, we evaluate the performance of $\hat{g}$ for $\sigma^* = 1/2$ which
<figure><img src="image_14.png" /><figcaption>Figure A.2: Risks of the plug-in classifier w.r.t values of the minimum gap $\theta$ between the drift functions</figcaption></figure>

is close enough to zero, and for larger value $\sigma^* \in \{1, 3/2\}$. We first consider the case where $\sigma^*$ is unknown. The results are given in Table A.5 and confirm our intuition. The error rate of the plug-in classifiers decreases as $\sigma^*$ decreases.

In a second step, we investigate the influence of estimating the coefficient $\sigma^*$ in the procedure. To evaluate this point, we assess the error rate of the plug-in classifier when $\sigma^* = 1$ is known. In this case, we only estimate the drift functions and the weights of mixture $pp^*$ to build our predictor. The results are given in Table A.6. First, we can notice that by comparison with results provided in Table A.5, there is almost no impact on the performance of the plug-in classifier when we assume the diffusion coefficient $\sigma^*$ in the Ornstein-Uhlenbeck model to be known or not.

Finally, we also study the influence of parameter $A_N$ on the estimation procedure. Indeed, our theoretical results indicates that $A_N$ should be of order $\sqrt{\log(N)}$ when $\sigma^*$ is constant and known, while $A_N = \log(N)$ is recommended when $\sigma^*$ is unknown. To this end, we evaluate the error rate of our procedure for these choices. The results are also provided in Table A.6 and show that the performance are almost the same in the two cases.

<table><caption>Table A.5: Evolution of the performance of the plug-in classifier $\hat{g}$ and of $g^*$ w.r.t values of the constant diffusion coefficient $\sigma^*$ for $N = 100$ and $n = 100$.</caption><thead><tr><th></th><th>$\hat{R}(\hat{g})$</th><th>$\hat{R}(g^*)$</th></tr></thead><tbody><tr><td>$\sigma^* = 1/2$</td><td>0.23 (0.04)</td><td>0.21 (0.01)</td></tr><tr><td>$\sigma^* = 1$</td><td>0.44 (0.05)</td><td>0.41 (0.01)</td></tr><tr><td>$\sigma^* = 3/2$</td><td>0.52 (0.05)</td><td>0.49 (0.01)</td></tr></tbody></table>
<table><caption>Table A.6: Risk classification of $\hat{g}$ when the diffusion $\sigma^* = 1$ is known, and $n = 100$.</caption><thead><tr><th></th><th>N = 100</th><th>N = 1000</th></tr></thead><tbody><tr><td>A<sub>N</sub> = &radic;log(N)</td><td>0.44 (0.05)</td><td>0.41 (0.05)</td></tr><tr><td>A<sub>N</sub> = log(N)</td><td>0.43 (0.05)</td><td>0.43 (0.05)</td></tr></tbody></table>

## A.6 Conclusion and discussion

In this paper, we propose a plug-in classifier for the multiclass classification of trajectories generated by a mixture of diffusion processes whose drift functions $b_i^*$, $i \in \mathcal{Y}$ and diffusion coefficient $\sigma^*$ are assumed to be unknown. In the considered model, each class $i$ is characterized by a drift function, $b_i^*$ whereas the diffusion coefficient $\sigma^*$ is common for all classes. This work extends to the nonparametric case, the multiclass classification procedure provided in [29] where $\sigma^* = 1$ and the drift functions depend on an unknown parameter $\theta \in \mathbb{R}^d$. Our proposed procedure relies on consistent projection estimators $\hat{b}_i$, $i \in \mathcal{Y}$ and $\hat{\sigma}^2$ of the drift and diffusion coefficients on a constrained approximation subspace spanned by the spline basis. We establish the consistency, w.r.t. the excess risk, of our procedure and then studied its rate of convergence under different kind of assumptions. In particular, we show that the proposed plug-in classifier reaches a rate of convergence of order $N^{-1/5}$ (up to a factor of order $\exp(\sqrt{c \log(N)})$) when $b^*$, $\sigma^*$, and $p^*$ are unknown. Besides, a numerical study illustrates the performance of our classification procedure.

In the case where $\sigma^* = 1$, we manage to derive faster rates of convergence. In particular, when the drift functions are bounded and Hölder with regularity $\beta \ge 1$, we obtained a rate of order $N^{-\beta/(2\beta+1)}$ (up to a factor of order $\exp(\sqrt{c \log(N)})$). Interestingly, this result can be viewed as an extension of the one obtained in [47] to the multiclass mixture model, where the drift functions are time-dependent. Furthermore, up to $\exp(\sqrt{c \log(N)})$ factor, our rate of convergence matches the optimal rates of convergence obtained in the univariate setting (e.g. $X \in \mathbb{R}$), in [4]. Finally, for the case of unbounded drift functions, we assume that the drift functions are the re-entrant. Taking advantage of this property, we establish that our plug-in classifier achieves a rate of convergence of order $N^{-3\beta/4(2\beta+1)}$. For $\beta = 1$, this rate of convergence is of the same order as the one obtained in [48] for plug-in classifier in the univariate classification setting, when the feature $X$ does not satisfy the strong density assumption.

A question that can be tackled for future research is the study of the optimality in the minimax sense of our plug-in procedure. In particular, the adaptivity of estimators of the drift and diffusion coefficients should be investigated. Furthermore, it might be interesting to consider the margin type assumption as in [47] to derive faster rates of convergence. Also, following [29], it is natural to derive theoretical properties for empirical risk minimization procedure based on convex losses. Finally, the extension to the high-dimensional setting would require further work. In particular, the control of the transition densities is different in this setting.

## A.7 Proofs

The section is devoted to the proofs of our main results. In order to simplify the notation, we write $\Delta_n = \Delta$. Besides, $C > 0$ is a constant which may change from one line to another. When the dependency on a parameter $\theta$ needs to be highlighted, we write $C_\theta$.

### A.7.1 Technical results on the process X

**Lemma A.7.1.** Under Assumption A.2.1 and for all integer $q \ge 1$, there exists $C^* > 0$ depending on $q$ such that for all $0 \le s < t \le 1$,

$$ \mathbb{E} |X_t - X_s|^2 \le C^*(t-s)^q. $$

The proof of Lemma A.7.1 is provided in Appendix.
For each $t \in [0, 1]$ and $x \in \mathbb{R}$, we denote by $p_X(t, x)$ the transition density of the underlying process $X_t$ given the starting point $X_0 = 0$. We also denote by $p_{i,X}(t, \cdot)$ the transition density of the process driven by the drift function $b_i^*$. Note that Assumption A.2.1 ensures the existence of the transition densities. The rest of this section is dedicated to some results on the transition densities $p_{i,X}$ for $i = 1, \dots, K$. Nonetheless, since the transition $p_X$ of the process $X$ writes as

$$p_X = \sum_{i=1}^{K} \mathfrak{p}_i^* p_{i,X},$$

all these results apply also for $p_X$. The following proposition is provided in [55] (Proposition 1.2).

**Proposition A.7.2.** Under Assumptions A.2.1 and A.2.2, there exist constants $c > 1$, $C > 1$ such that for all $t \in (0, 1]$, $x \in \mathbb{R}$, and $i = 1, \dots, K$

$$\frac{1}{C\sqrt{t}} \exp\left(-c\frac{x^2}{t}\right) \le p_{i,X}(t, x) \le \frac{C}{\sqrt{t}} \exp\left(-\frac{x^2}{ct}\right).$$

From this result, we can deduce an evaluation of the probability of the process to exit a compact set. This is the purpose of the next result.

**Lemma A.7.3.** Under Assumption A.2.1 and A.2.2, there exist $C_1, C_2 > 0$ such that for all $A > 0$

$$\sup_{t \in [0,1]} \mathbb{P}(|X_t| \ge A) \le \frac{C_1}{A} \exp(-C_2 A^2).$$

*Proof.* Let $A > 0$, we have for $t \in (0, 1]$,

$$\mathbb{P}(|X_t| \ge A) = \int_A^{+\infty} p_X(t, x)dx + \int_A^{+\infty} p_X(t, -x)dx.$$

From Proposition A.7.2, we then deduce that

$$\mathbb{P}(|X_t| \ge A) \le C \frac{\sqrt{t}}{A} \int_A^{+\infty} c \frac{2x}{t} \exp(-c \frac{x^2}{t}) dx \le C \frac{\sqrt{t}}{A} \exp(-c \frac{A^2}{t}).$$

From the above inequality, and using that $t \in (0, 1]$, we deduce the result. $\square$

**Lemma A.7.4.** Under Assumption A.2.1, there exist $C_0, C_1$, and $C_2$, such that for $i = 1, \dots, K$, for $x \in [-A, A]$, we have

$$C_1 \exp(-C_2 A^2) \le \frac{1}{n} \sum_{k=1}^{n-1} p_{i,X}(k\Delta, x) \le C_0.$$

*Proof of Lemma A.7.4.* For $i \in \{1, \dots, K\}$, for all $x \in \mathbb{R}$, we have from Proposition A.7.2,

$$\frac{1}{n} \sum_{k=1}^{n} p_{i,X}(k\Delta, x) \le \frac{C}{n} \sum_{k=1}^{n} \frac{1}{\sqrt{k\Delta}} = \frac{C}{\sqrt{n}} \sum_{k=1}^{n} \frac{1}{\sqrt{k}} \le \frac{2C}{\sqrt{n}} \sum_{k=1}^{n} \frac{1}{\sqrt{k+1}} \quad (A.19)$$

Since the function $x \mapsto \frac{1}{\sqrt{x}}$ is decreasing over $[1, +\infty[$, we deduce from Equation (A.19) that

$$\frac{1}{n} \sum_{k=1}^{n} p_{i,X}(k\Delta, x) \le \frac{4C\sqrt{n+1}}{\sqrt{n}} \le C_0,$$

which gives the upper bound. For the lower bound, we observe from Proposition A.7.2 that for $k \in [1, n-1]$, and $x \in \mathbb{R}$,

$$C \exp\left(-\frac{cx^2}{k\Delta}\right) \le \frac{C}{\sqrt{k\Delta}} \exp\left(-\frac{cx^2}{k\Delta}\right) \le p_i(k\Delta, x). \quad (A.20)$$
Since $g : (s, x) \mapsto \exp(-\frac{cx^2}{s})$ is strictly increasing in $s$ over $[0, 1]$, we obtain for $k \in [1, n-1]$,

$$ \int_{\frac{1}{n}}^{\frac{n-1}{n}} g(s,x)ds \le \sum_{k=2}^{n-1} \int_{(k-1)\Delta}^{k\Delta} (g(k\Delta, x) + g(s,x) - g(k\Delta, x)) ds \le \frac{1}{n} \sum_{k=1}^{n-1} \exp\left(-\frac{cx^2}{k\Delta}\right). $$

Hence, we deduce that for $n \ge 3$, and $x \in [-A, A]$

$$ \frac{1}{6} \exp(-2cA^2) \le \int_{\frac{1}{2}}^{\frac{n-1}{n}} g(s,x)ds \le \frac{1}{n} \sum_{k=1}^{n-1} \exp\left(-\frac{cx^2}{k\Delta}\right). $$

For the first lower bound, we use that $g(s, x) \ge e^{-2cA^2}$ for $x \in [-A, A]$ and $s \ge 1/2$, and that the length of $[1/2, (n-1)/n]$ is larger than $1/6$ for $n \ge 3$. This explains our choice of integration interval in the middle term of the above inequalities. Finally, gathering this bound with Equation (A.20), leads to

$$ \frac{1}{6} \exp(-2cA^2) \le \frac{1}{n} \sum_{k=1}^{n-1} \exp\left(-\frac{cx^2}{k\Delta}\right) \le \frac{1}{n} \sum_{k=1}^{n-1} p_{i,X}(k\Delta, x). $$

□

**Lemma A.7.5.** Suppose that $\sigma^*$ is a constant. Under Assumption A.2.1, and for all $q > 1$, there exists $K_q > 1$ such that for all $(t, x) \in (0, 1] \times [-A, A]$,

$$ \frac{1}{K_q \sqrt{t}} \exp\left(-\frac{2q-1}{2q\sigma^{*2}t} x^2\right) \le p_X(t, x) \le \frac{K_q}{\sqrt{t}} \exp\left(-\frac{x^2}{2q\sigma^{*2}t}\right). $$

**Proof of Lemma A.7.5.** The transition density $p_X^0$ of the process $(0 + \sigma^* W_t)_{t \in [0,1]}$ (with a constant diffusion coefficient $\sigma^*$) is given by

$$ p_X^0(t, x) := \frac{1}{\sqrt{2\pi\sigma^{*2}t}} \exp\left(-\frac{1}{2\sigma^{*2}t}|0-x|^2\right). \quad (A.21) $$

We are going to demonstrate the inequality for $p_{i,X}$, which is the transition density of $X$ in class number $i$. Indeed, then it will be true for all $i \in \mathcal{Y}$ and thus for $p_X = p_{Y,X}$. We follow here the arguments given in the *proof of (1.6)* in [55]. Let us denote,

$$ Z_{i,t} = \exp \left( \int_{0}^{t} \frac{b_{i}^{*}(X_{s})}{\sigma^{*}} dW_{s} - \int_{0}^{t} \frac{b_{i}^{*2}(X_{s})}{\sigma^{*2}} ds \right). $$

We have $\forall (t, x) \in [0, 1] \times \mathbb{R}$,

$$ p_{i,X}(t, x) = p_X^0(t, x) \mathbb{E}^0 [Z_{i,t} | X_t = x], $$

and

$$ \frac{1}{p_{i,X}(t, x)} \le \frac{1}{p_X^0(t, x)} \mathbb{E}^0 [Z_{i,t}^{-1} | X_t = x]. \quad (A.22) $$

Then,

$$ \mathbb{E}^0 [Z_{i,t} | X_t = x] = 1 + \frac{1}{p_X^0(t, x)} \int_0^t \mathbb{E}^0 \left[ Z_{i,t} b_i^*(X_s) \frac{X_s - x}{\sigma^{*2}(t-s)} p_X^0(t-s, x) \right] ds $$

and

$$ \mathbb{E}^0 [Z_{i,t}^{-1} | X_t = x] = 1 + \frac{1}{p_X^0(t, x)} \int_0^t \mathbb{E}^0 \left[ Z_{i,s}^{-1} b_i^*(X_s) \frac{X_s - x}{\sigma^{*2}(t-s)} p_X^0(t-s, x) \right] ds. $$
For all $(t, x) \in ]0, 1] \times \mathbb{R}$, one has :

$$
\begin{aligned}
\mathbb{E}^0 [Z_{i,t} | X_t = x] &= 1 + \frac{1}{p_X^0(t, x)} \int_0^t \mathbb{E}^0 \left[ Z_{i,s} b_i^*(X_s) \frac{X_s - x}{\sigma^{*2}(t-s)} p_X^0(t-s, x) \right] ds \\
&\le 1 + \frac{C}{p_X^0(t, x)} \int_0^t \mathbb{E}^0 \left[ Z_{i,s} |b_i^*(X_s)| \frac{|X_s - x|}{(t-s)^{3/2}} \exp \left( -\frac{(X_s - x)^2}{2\sigma^{*2}(t-s)} \right) \right] ds \\
&\le 1 + \frac{C}{p_X^0(t, x)} \int_0^t \mathbb{E}^0 \left[ Z_{i,s} |b_i^*(X_s)| \frac{1}{\varepsilon(t-s)} \exp \left( -\frac{(1-\varepsilon)(X_s - x)^2}{2\sigma^{*2}(t-s)} \right) \right] ds,
\end{aligned}
$$

using that $y\varepsilon \exp(-\varepsilon y^2/2) \le 1$ for $0 < \varepsilon < 1$. Let $q, q' > 1$ be two real numbers such that $\frac{1}{q} + \frac{1}{q'} = 1$. Using Hölder's inequality, and the Lipschitz property of $b^*$, one has:

$$
\mathbb{E}^0 [Z_{i,t} | X_t = x] \le 1 + \frac{C \varepsilon^{-1}}{p_X^0(t, x)} \int_0^t \left( \mathbb{E}^0 \left[ \frac{Z_{i,s}^q q (1 + |X_s|)^q}{(t-s)^q} \right] \right)^{\frac{1}{q}} \left( \mathbb{E}^0 \left[ \exp \left( -\frac{(X_s - x)^2}{2\sigma^{*2}(t-s)} \right) \right] \right)^{\frac{1}{q'}} ds, \quad (A.23)
$$

with $\varepsilon = 1 - 1/q'$. According to Lemma A.1 in [55], one has:

$$
\forall q > 1, \quad \mathbb{E}^0 \left[ Z_{i,s}^q (1 + |X_s|)^q \right] + \mathbb{E}^0 \left[ Z_{i,s}^{-q} (1 + |X_s|)^q \right] \le C_1,
$$

where $C_1 > 0$ is a constant. Thus, it remains to upper bound $\mathbb{E}^0 \left[ \exp \left( -\frac{(X_s - x)^2}{2\sigma^{*2}(t-s)} \right) \right]$ and then deduce an upper bound of $\mathbb{E}^0 [Z_{i,t} | X_t = x]$. For all $s < t$, we have:

$$
\sqrt{2\pi\sigma^{*2}s} \mathbb{E}^0 \left[ \exp \left( -\frac{(X_s - x)^2}{2\sigma^{*2}(t-s)} \right) \right] = \int_{\mathbb{R}} \exp \left( -\frac{1}{2\sigma^{*2}(t-s)} (z-x)^2 \right) \exp \left( -\frac{1}{2\sigma^{*2}s} z^2 \right) dz.
$$

It follows that,

$$
\mathbb{E}^0 \left[ \exp \left( - \frac{(X_s - x)^2}{2\sigma^{*2}(t-s)} \right) \right] = \sqrt{\frac{t-s}{t}} \exp \left( - \frac{x^2}{2\sigma^{*2}t} \right).
$$

Thus, from Equation (A.23), we obtain:

$$
\begin{aligned}
\mathbb{E}^0 [Z_{i,t} | X_t = x] &\le 1 + \frac{C \varepsilon^{-1}}{p_X^0(t, x)} \int_0^t \frac{(t-s)^{\frac{1}{2q'}-1}}{t^{\frac{1}{2q'}}} \exp \left( -\frac{x^2}{2q' \sigma^{*2} t} \right) ds \\
&\le 1 + \frac{C \varepsilon^{-1} t^{-1/2q'}}{p_X^0(t, x)} \exp \left( -\frac{x^2}{2q' \sigma^{*2} t} \right) [-2q' (t-s)^{1/2q'}]_0^t \\
&\le 1 + \frac{C \varepsilon^{-1}}{p_X^0(t, x) \sqrt{t}} \exp \left( -\frac{x^2}{2q' \sigma^{*2} t} \right),
\end{aligned}
$$

From the definition of function $p_X^0$ given in Equation (A.21) together with relation (A.22), we obtain that

$$
p_{i,X}(t, x) \le p_X^0(t, x) \left( 1 + \frac{C \varepsilon^{-1}}{p_X^0(t, x) \sqrt{t}} \exp \left( - \frac{x^2}{2q' \sigma^{*2} t} \right) \right).
$$

Thus, there exists a constant $K_q > 1$ (as $\varepsilon = 1 - 1/q'$ and $1/q + 1/q' = 1$) such that,

$$
\forall (t, x) \in ]0, 1] \times \mathbb{R}, \quad p_{i,X}(t, x) \le \frac{K_q}{\sqrt{t}} \exp \left( -\frac{x^2}{2q' \sigma^{*2} t} \right), \quad \forall q' > 1. \quad (A.24)
$$

Following the same lines, one has

$$
\mathbb{E}^0 [Z_{i,t}^{-1} | X_t = x] \le 1 + \frac{C^{\text{te}}}{p_X^0(t, x) \sqrt{t}} \exp \left( -\frac{x^2}{2q' \sigma^{*2} t} \right).
$$

Also, there exists a constant $K_q > 1$, such that,

$$
\forall (t, x) \in ]0, 1] \times \mathbb{R} \ p_{i,X}(t, x) \ge \frac{1}{K_q \sqrt{t}} \exp \left( -\frac{2q' - 1}{2q' \sigma^{*2} t} x^2 \right), \quad \forall q' > 1. \quad (A.25)
$$

The final result is deduced from (A.24) and (A.25). $\square$
### A.7.2 Proofs of Section A.3

We begin by providing the proof of Theorem A.3.2 that relies in part on Proposition A.3.1.

**Proof of Theorem A.3.2.** From Proposition A.3.1, we have the following inequality

$$ \mathbb{E}[\mathcal{R}(\hat{g}) - \mathcal{R}(g^*)] \le 2 \sum_{i=1}^{K} \mathbb{E}[|\hat{\pi}_i(X) - \pi_i^*(X)|]. \quad (A.26) $$

We define $\bar{\mathbf{F}}$ the discretized version of $\mathbf{F}^*$,

$$ \bar{\mathbf{F}} = (\bar{F}_1, \dots, \bar{F}_K), \quad \text{with} \quad \bar{F}_i(X) = \sum_{k=0}^{n-1} \left( \frac{b_i^*}{\sigma^{*2}}(X_{k\Delta}) (X_{(k+1)\Delta} - X_{k\Delta}) - \frac{\Delta}{2} \frac{b_i^{*2}}{\sigma^{*2}}(X_{k\Delta}) \right), $$

and for each $i \in \mathcal{Y}$, $\bar{\pi}_i^* = \phi_i(\bar{\mathbf{F}})$ the discretized version of $\pi_i^*$, and $\bar{\pi}_i = \phi_i(\hat{\mathbf{F}})$. From Equation (A.26), we deduce

$$ \begin{aligned} \mathbb{E}[\mathcal{R}(\hat{g}) - \mathcal{R}(g^*)] &\le 2 \left( \sum_{i=1}^{K} \mathbb{E}[|\hat{\pi}_i(X) - \bar{\pi}_i(X)|] + \mathbb{E}[|\bar{\pi}_i(X) - \bar{\pi}_i^*(X)|] \right. \\ &\qquad \left. + \sum_{i=1}^{K} \mathbb{E}[|\bar{\pi}_i^*(X) - \pi_i^*(X)|] \right) \\ &\le 2 \sum_{i=1}^{K} \mathbb{E}[|\hat{\phi}_i(\hat{\mathbf{F}}(X)) - \phi_i(\hat{\mathbf{F}}(X))|] + 2 \sum_{i=1}^{K} \mathbb{E}[|\phi_i(\hat{\mathbf{F}}(X)) - \phi_i(\bar{\mathbf{F}}(X))|] \\ &\qquad + 2 \sum_{i=1}^{K} \mathbb{E}[|\phi_i(\bar{\mathbf{F}}(X)) - \phi_i(\mathbf{F}^*(X))|]. \end{aligned} \quad (A.27) $$

For the first term of the r.h.s. of the above inequality, we observe that for $(x_1, \dots, x_K) \in \mathbb{R}^K$, and $(i, j) \in \mathcal{Y}^2$ we have

$$ \left| \frac{\partial}{\partial \mathfrak{p}_j^*} \frac{\mathfrak{p}_i^* \exp(x_i)}{\sum_{k=1}^K \mathfrak{p}_k^* \exp(x_k)} \right| \le \frac{1}{\mathfrak{p}_0^*}. $$

Therefore,

$$ \sum_{i=1}^{K} \mathbb{E} \left[ \left| \hat{\phi}_i(\hat{\mathbf{F}}(X)) - \phi_i(\hat{\mathbf{F}}(X)) \right| \right] \le C_{K, \mathfrak{p}_0^*} \sum_{k=1}^{K} \mathbb{E} \left[ \left| \hat{\mathfrak{p}}_k - \mathfrak{p}_k \right| \right] \le \frac{C_{K, \mathfrak{p}_0^*}}{\sqrt{N}}. \quad (A.28) $$

For the second term of Equation (A.27), since the softmax function is 1-Lipschitz, we have for $j \in \mathcal{Y}$

$$ \mathbb{E} \left| \phi_j(\hat{\mathbf{F}}(X)) - \phi_j(\bar{\mathbf{F}}(X)) \right| \le \sum_{i=1}^{K} \mathbb{E} \left| \left| \hat{F}_i(X) - \bar{F}_i(X) \right| \right|. $$

We set $\xi(s) := k\Delta$, if $s \in [k\Delta, (k+1)\Delta)$, for $k \in [0, n-1]$. We then deduce that

$$ \left| \hat{F}_i(X) - \bar{F}_i(X) \right| \le \int_0^1 \left| \left( \frac{\hat{b}_i}{\hat{\sigma}^2} - \frac{b_i^*}{\sigma^{*2}} \right) (X_{\xi(s)}) b_Y^*(X_s) \right| ds + \frac{1}{2} \int_0^1 \left| \left( \frac{\hat{b}_i^2}{\hat{\sigma}^2} - \frac{b_i^{*2}}{\sigma^{*2}} \right) (X_{\xi(s)}) \right| ds \\ + \left| \int_0^1 \left( \frac{\hat{b}_i}{\hat{\sigma}^2} - \frac{b_i^*}{\sigma^{*2}} \right) (X_{\xi(s)}) \sigma^*(X_s) dW_s \right|, $$

which implies

$$ \mathbb{E} \left[ \left| \hat{F}_i(X) - \bar{F}_i(X) \right| \right] \le \mathbb{E} \left[ \int_0^1 \left| \left( \frac{\hat{b}_i}{\hat{\sigma}^2} - \frac{b_i^*}{\sigma^{*2}} \right) (X_{\xi(s)}) b_Y^*(X_s) \right| ds \right] + \frac{1}{2} \mathbb{E} \left[ \int_0^1 \left| \left( \frac{\hat{b}_i^2}{\hat{\sigma}^2} - \frac{b_i^{*2}}{\sigma^{*2}} \right) (X_{\xi(s)}) \right| ds \right] \\ + \mathbb{E} \left[ \int_0^1 \left( \frac{\hat{b}_i}{\hat{\sigma}^2} - \frac{b_i^*}{\sigma^{*2}} \right)^2 (X_{\xi(s)}) \sigma^{*2}(X_s) ds \right]. $$
Since for all $x, \sigma^*(x) \ge \sigma_0^*, \text{ and } \hat{\sigma} \ge \sigma_0$, we get

$$ \left\{ \begin{array}{l} \left| \frac{\hat{b}_i}{\hat{\sigma}^2}(x) - \frac{b_i^*}{\sigma^{*2}}(x) \right| \le \sigma_0^{-2} \left| \hat{b}_i(x) - b_i^*(x) \right| + \sigma_0^{-2} \sigma_0^{*-2} |b_i^*(x)| |\hat{\sigma}^2(x) - \sigma^{*2}(x)|, \\ \\ \left| \frac{\hat{b}_i^2}{\hat{\sigma}^2}(x) - \frac{b_i^{*2}}{\sigma^{*2}}(x) \right| \le \sigma_0^{-2} \left| \hat{b}_i(x) + b_i^*(x) \right| \left| \hat{b}_i(x) - b_i^*(x) \right| + \sigma_0^{-2} \sigma_0^{*-2} |b_i^*(x)|^2 |\hat{\sigma}^2(x) - \sigma^{*2}(x)|. \end{array} \right. \quad (A.29) $$

Hence, as $\hat{b}_i(x) \le b_{\max}$, and $\mathbb{E}[\sup_{t \in [0,1]} |b_i^*(X_t)|] \le C_1$, the above inequalities and the Cauchy-Schwarz inequality yield

$$ \mathbb{E} \left| \hat{F}_i(X) - \bar{F}_i(X) \right| \le C_{\sigma_0^*} \sigma_0^{-2} \left( b_{\max} \mathbb{E} \left\| \hat{b}_i - b_i^* \right\|_n + \mathbb{E} \left\| \hat{\sigma}^2 - \sigma^2 \right\|_n \right). $$

Therefore, we have,

$$ \sum_{i=1}^{K} \mathbb{E} \left[ \left| \phi_i(\hat{F}(X)) - \phi_i(\bar{F}(X)) \right| \right] \le C_{K, \sigma_0^*} \sigma_0^{-2} \sum_{i=1}^{K} \left( b_{\max} \mathbb{E} \left\| \hat{b}_i - b_i^* \right\|_n + \mathbb{E} \left\| \hat{\sigma}^2 - \sigma^2 \right\|_n \right). \quad (A.30) $$

Finally, the last term is bounded as follows. We first observe that for all $i \in \mathcal{Y}$

$$ \begin{aligned} \mathbb{E} \left[ |\bar{F}_i(X) - F_i^*(X)|^2 \right] &\le 3 \mathbb{E} \int_0^1 \left( \frac{b_i^*(X_{\xi(s)})}{\sigma^{*2}(X_{\xi(s)})} - \frac{b_i^*(X_s)}{\sigma^{*2}(X_s)} \right)^2 b_Y^{*2}(X_s) ds \\ &\quad + 3 \mathbb{E} \int_0^1 \left( \frac{b_i^{*2}(X_{\xi(s)})}{\sigma^{*2}(X_{\xi(s)})} - \frac{b_i^{*2}(X_s)}{\sigma^{*2}(X_s)} \right)^2 ds + 3 \mathbb{E} \int_0^1 \left( \frac{b_i^*(X_{\xi(s)})}{\sigma^{*2}(X_{\xi(s)})} - \frac{b_i^*(X_s)}{\sigma^{*2}(X_s)} \right)^2 \sigma^{*2}(X_s) ds. \end{aligned} $$

Using again that $\sigma^*(\cdot) \ge \sigma_0^*$, and $\mathbb{E}[\sup_{t \in [0,1]} |b_i^*(X_t)|^q] \le C$ for $q \ge 1$ (by Assumption A.2.1), the Cauchy-Schwarz inequality implies

$$ \begin{aligned} \mathbb{E} \left[ |\bar{F}_i(X) - F_i^*(X)|^2 \right] &\le C_{\sigma_0^*} \left( \int_0^1 \mathbb{E} \left[ |b_i^*(X_{\xi(s)}) - b_i^*(X_s)|^2 \right] ds \right. \\ &\quad \left. + \int_0^1 \sqrt{\mathbb{E} \left[ |b_i^*(X_{\xi(s)}) - b_i^*(X_s)|^4 \right]} ds + \int_0^1 \sqrt{\mathbb{E} \left[ |\sigma^{*2}(X_{\xi(s)}) - \sigma^{*2}(X_s)|^4 \right]} ds \right). \end{aligned} $$

Finally, since the functions $b_i^*$, and $\sigma^*$ are Lipschitz, we deduce from Lemma A.7.1 that

$$ \mathbb{E} \left[ |\bar{F}_i(X) - F_i^*(X)|^2 \right] \le C_{\sigma_0^*} \Delta, $$

which implies together with the fact that the soffmax function is 1-Lipschitz and the Jensen inequality that

$$ \sum_{i=1}^{K} \mathbb{E} \left[ |\phi_i(\bar{F}(X)) - \phi_i(\bar{F}^*(X))| \right] \le C_{K, \sigma_0^*} \sqrt{\Delta}. \quad (A.31) $$

In view of Equation A.27, the combination of Equations (A.28), and (A.30), and (A.31) yields the desired result. $\square$

**Proof of Proposition A.3.3.** We consider $h$ a $L$-Lipschitz function. We define the spline-approximation $\tilde{h}$ of $h$ by

$$ \tilde{h}(x) := \sum_{\ell=-M}^{K^*-1} h(u_\ell) B_\ell(x), \quad \forall x \in \mathbb{R}. $$

First, we note that $\tilde{h} \in S_{K^*,M}$. Indeed, since $h$ is $L$-Lipschitz, there exists $C_L > 0$ such that for $A$ large enough,

$$ |h(x)| \le C_L(1 + |x|) \le CA, \quad \forall x \in [-A, A]. $$
Therefore, for $N$ large enough, we have

$$|h(x)| \le A \log^{1/2}(N).$$

Then, we deduce

$$\sum_{\ell=-M}^{\Xi-1} h^2(u_\ell) \le (\Xi + M)A^2 \log(N).$$

For $x \in [-A, A]$, there exists $0 \le \ell_0 \le \Xi - 1$ such that $x \in [u_{\ell_0}, u_{\ell_0+1})$. We use the following property of the B-spline basis

$$B_{\ell}(x) = 0, \text{ if } x \notin [u_{\ell}, u_{\ell+M+1}), \quad \ell = -M, \dots, K_N + M.$$

Hence, for $x \in [u_{\ell_0}, u_{\ell_0+1})$, we have $B_{\ell}(x) = 0$ for $\ell \le \ell_0 - M - 1$, and $\ell \ge \ell_0 + M$. Thus,

$$\begin{aligned}
|\tilde{h}(x) - h(x)| & \le \sum_{\ell=-M}^{\Xi-1} |h(u_\ell) - h(x)| B_\ell(x) \\
& = \sum_{\ell=\ell_0-M}^{\ell_0} |h(u_\ell) - h(x)| B_\ell(x) \\
& \le \max_{\ell=\ell_0-M, \dots, \ell_0} |h(u_\ell) - h(x)| \\
& \le L(u_{\ell_0+1} - u_{\ell_0-M}) \le \frac{2L(M+1)A}{\Xi},
\end{aligned}$$

which concludes the proof.

<figure><img src="image_15.png" /></figure>

**Proof of Theorem A.3.4.** The proof is divided in two parts. The first part establishes the rates of convergence of the drift estimators, and the second part is devoted to the study of the rates of convergence of the diffusion coefficient estimator.

**Rates of convergence for drift estimator.** Let $i \in \{1, \dots, K\}$. We introduce, on the random event $\{N_i > 1\}$, and with $A_{N_i} = \log(N_i)$, the function,

$$\bar{b}_i := b_i^* \mathbb{1}_{(-A_{N_i}, A_{N_i})}.$$

We recall that $N_i = \sum_{j=1}^N \mathbb{1}_{\{Y_j=i\}}$ is the random number of paths in the class number $i$. For a function $h$, we introduce the empirical norm of class $i$ on the event $\{N_i > 1\}$ as

$$\|h\|_{n,N_i}^2 := \frac{1}{nN_i} \sum_{j \in \mathcal{I}_i} \sum_{k=0}^{n-1} h^2(X_{k\Delta}^j).$$

We first observe that

$$\mathbb{E} \left[ \left\| \hat{b}_i - b_i^* \right\|_n \right] = \mathbb{E} \left[ \left\| \hat{b}_i - b_i^* \right\|_n \mathbb{1}_{\{N_i > 1\}} \right] + \mathbb{E} \left[ \left\| \hat{b}_i - b_i^* \right\|_n \mathbb{1}_{\{N_i \le 1\}} \right].$$

Let us work at first on the event $\{N_i > 1\}$. For all $i \in \mathcal{Y}$, we define the following conditional expectation

$$\mathbb{E}_i[\cdot] = \mathbb{E}[\cdot | \mathbb{1}_{\{Y_1=i\}}, \dots, \mathbb{1}_{\{Y_N=i\}}].$$

We apply Proposition A.3.3, and Proposition 3.2 of [30] on the event $\{N_i > 1\}$ and deduce that

$$\mathbb{E}_i \left[ \left\| \hat{b}_i - \bar{b}_i \right\|_{n,N_i}^2 \right] \le C \left( \frac{A_{N_i}^2}{K_{N_i}^2} + \sqrt{\frac{K_{N_i} A_{N_i}^2 \log(N)}{N_i}} + \Delta \right). \quad (A.32)$$