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# Nonparametric drift estimation for i.i.d. paths of censored stochastic differential equations (SDE)

Yichuan Huang

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Yichuan Huang. Nonparametric drift estimation for i.i.d. paths of censored stochastic differential equations (SDE). 2025. hal-05307782

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# Nonparametric drift estimation for i.i.d. paths of censored stochastic differential equations (SDE)

Yichuan Huang<fnref n="1" />,<fnref n="2" />

**Abstract.** We study nonparametric drift estimation for a diffusion observed under continuous-time right censoring. The latent process $X$ solves $dX_t = b(X_t) dt + \sigma(X_t) dW_t$, while the observer records $Y_t = \min(X_t, C_t)$ together with the censoring indicator $\delta_t = \mathbf{1}_{\{X_t < C_t\}}$, where $C$ is a continuous semimartingale independent of $X$. Based on $N$ i.i.d. trajectories over a fixed time horizon $[0, T]$, we construct a problem-specific contrast and a projection least-squares estimator of $b$ on approximation spaces. We establish nonasymptotic risk bounds, propose an adaptive dimension selection rule with explicit penalties, and derive convergence rates over Besov balls on compact domains and Sobolev-type classes on non-compact domains. We also introduce a diagnostic procedure to assess the level of censoring and illustrate, through simulations, how censoring mechanisms affect estimation performance.

## 1. Introduction

Many continuous-time measurements are right-censored: once the signal enters a region above the device’s admissible range, the readout is capped at an upper limit, and the exact value is not observed. A classical example arises in survival and reliability analysis: at the end of the follow-up, some lifetimes are only known to exceed the last observation time Kaplan and Meier (1958); Cox (1972); Lawless (2011). Measuring devices can also reach their maximum capacity: air-quality or radiation monitors often impose the highest reportable value, leading to level-censoring at the top of the scale Helsel (2011). In finance, continuously monitored barrier mechanisms effectively limit the evolution of an underlying price once the barrier is reached Merton et al. (1971); Hull et al. (2009). In this paper, we study diffusion signals observed under continuous-time right censoring and show how to recover drift features from such capped trajectories.

We model the signal as a one-dimensional Itô diffusion

$$dX_t = b(X_t) dt + \sigma(X_t) dW_t, \quad X_0 = x_0, \qquad (1.1)$$

observed on $[0, T]$ through a continuous-time right-censoring device:

$$Y_t := X_t \wedge C_t, \quad \delta_t := \mathbf{1}_{\{X_t < C_t\}}, \quad 0 \le t \le T,$$

where $C$ is a continuous semimartingale independent of $X$. The sample consists of $N$ i.i.d. copies $\{(Y^i, \delta^i)\}_{i=1}^N$ of $(Y, \delta)$. This formulation captures the experimental limitations outlined above: continuous monitoring, hard bounds, and on/off regimes mapped by $(\delta_t)$. In the complete-observation

<footnote n="1">¹ Laboratoire Modal’X, Université Paris Nanterre, Nanterre, France.</footnote>

<footnote n="2">² Laboratoire MAP5, Université Paris Cité.</footnote>

<footnote>E-mail address: yhuang@parisnanterre.fr</footnote>

<footnote>URL: https://sites.google.com/view/yichuanhuang</footnote>

<footnote>This research was partially supported by the Laboratoire d'Excellence (Labex) MME-DII (ANR-11-LBX-0023) under the “Investissements d’Avenir” program.</footnote>
case ($\delta_t \equiv 1$; equivalently $C_t \equiv +\infty$), following Marie (2025, Page 4), under the regime of continuous observation (over multiple trajectories), the estimation of the diffusion coefficient $\sigma$ is not a statistical problem: $\sigma$ is pointwise identifiable from the quadratic variation of a single path. Under right censoring, $\sigma$ remains identifiable only in the set of states that are visited while uncensored; see Lemma 2.2. Motivated by this, the present work treats $\sigma$ as (pathwise) recoverable and focuses on the nonparametric estimation of the drift function b.

Nonparametric drift estimation has traditionally been studied in two regimes: (a) long-time, discretely sampled single trajectories under the ergodicity of (1.1); see, e.g., Hoffmann (1999); Comte et al. (2007); Comte and Genon-Catalot (2021); Nickl and Ray (2020); and (b) independent trajectories observed over a fixed time horizon, leveraging cross-path independence; see, e.g., Comte and Genon-Catalot (2020a); Marie and Rosier (2023); Marie (2025). The second regime has been investigated more recently, and this setting is considered in our paper.

In the fully observed case, Comte and Genon-Catalot (2020a) analyzes a projection least-squares (pLS) estimator of b obtained by minimizing, over finite-dimensional spaces $S_m$, the standard (uncensored) contrast

$$ \tilde{\gamma}_N(h) := \frac{1}{NT} \sum_{i=1}^{N} \int_{0}^{T} h^2(X_t^i) dt - \frac{2}{NT} \sum_{i=1}^{N} \int_{0}^{T} h(X_t^i) dX_t^i, $$

which has been proven to be theoretically efficient and numerically stable. However, Comte and Genon-Catalot (2020a) assumes fully observed trajectories, and $\tilde{\gamma}_N$ is not observable under censoring.

In our setting, we replace $X$ with the observed process $Y = X \wedge C$, and we also consider the censoring indicator $\delta = 1_{\{X<C\}}$, leading to the survival-weighted contrast

$$ \gamma_N(h) := \frac{1}{NT} \sum_{i=1}^{N} \int_{0}^{T} \delta_t^i h^2(Y_t^i) dt - \frac{2}{NT} \sum_{i=1}^{N} \int_{0}^{T} \delta_t^i h(Y_t^i) dY_t^i, $$

which allows us to take into account the lack of information due to censoring. Section 3 details the construction, stabilization (truncation), and risk control.

Right-censoring is central to survival analysis: the event of interest (e.g., failure or death) may not occur before the follow-up ends. Thus, for some subjects, we only know that their lifetime exceeds a censoring time. The statistical aim is to recover features of the lifetime distribution—such as the survival function, the hazard rate, or the density—from this incomplete information. This problem has generated a broad range of nonparametric methods: the Kaplan-Meier estimator of the survival curve Kaplan and Meier (1958); cumulative-hazard estimators and the counting-process formulation Fleming and Harrington (2013); and regression models for censored outcomes, ranging from the Cox semiparametric hazard regression Cox (1972) to other nonparametric and semiparametric alternatives Buckley and James (1979); Beran (1981). Beyond these classical approaches, modern work develops adaptive procedures with rate-optimal guarantees for censored samples in density estimation Efromovich (2001, 2024). A comprehensive synthesis and unification of these developments is provided in Efromovich's monograph Efromovich (2025).

In the nonparametric regression framework with censored data—for example, in Zheng (1988); Fan and Gijbels (1994); Park (2004)—we consider i.i.d. observations

$$ (Y_i^*, \delta_i^*, X_i^*), \quad i = 1, \dots, n, $$
where

$$Y_i^* = \min(Z_i^*, C_i^*), \quad \delta_i^* = \mathbf{1}_{\{Z_i^* \le C_i^*\}},$$

$Z_i^*$ is the response of interest, $C_i$ is the censoring time, and $X_i^*$ is a covariate. The goal is to estimate the regression function

$$m(x) = \mathbb{E}[Z^* | X^* = x].$$

This problem can be addressed, for instance, by pLS estimators; see Brunel and Comte (2006).

Methodologically, the nonparametric inference for the process $X_t$ in (1.1) can be viewed as a variant of nonparametric regression (see, for example, Marie, 2025). However, when $X_t$ is censored, our setting departs from the existing literature in two essential respects. First, in the works cited above, the response $Z_i^*$ is right-censored at a single time, while the covariate $X_i^*$ is fully observed; in our model, the process $X_t$ (which may be seen as the covariate) is continuously censored along its trajectory. Second, the observable increments $dY_t$ (which may be regarded as the response) inherit this censoring structure. Estimation with a censored response is comparatively less developed; existing approaches typically rely on weighting by estimated conditional hazards (see, e.g., Jiang et al., 2022). Our estimation strategy is fundamentally different—we develop a pLS approach specifically tailored to the censored diffusion structure.

To the best of our knowledge, this paper is the first to address nonparametric drift estimation for diffusion processes observed through independent copies under continuous-time right censoring. Our contributions are as follows:

(1) We formalize the observation model $(Y, \delta)$ generated by a diffusion $(X_t)$ and an independent continuous semimartingale censoring process $(C_t)$, and we introduce a survival-weighted design measure that naturally appears in the risk analysis.

(2) We construct a pLS estimator of $b$ on approximation spaces $S_m$ spanned by bases suited to the domain: Laguerre on $\mathbb{R}_+$, Hermite on $\mathbb{R}$, and trigonometric (Fourier) on compact intervals $A \subset \mathbb{R}$. The construction exploits censoring indicators and survival weights to correct the bias induced by censoring.

(3) For any fixed $m$ that satisfies a stability condition (specified later), we establish a nonasymptotic bias-variance bound for the truncated estimator. When $\sigma$ is bounded on $A$, the variance term is controlled linearly in $m$.

(4) We propose a data-driven penalized selection of the dimension $m$ and derive nonasymptotic adaptive oracle inequalities. The penalty is explicit under bounded $\sigma$ and admits a computable modification in the unbounded case.

The paper is organized as follows. Section 2 introduces the diffusion model, the censoring mechanism, and the standing assumptions. Section 3 presents the projection least-squares estimator along with its stabilized variant. Section 4 establishes risk bounds, convergence rates, and the adaptive dimension selection procedure. Section 5 reports simulation results for several censoring scenarios. Section 6 offers concluding remarks and perspectives for future research. Finally, proofs and auxiliary materials are collected in Section 7 and the appendices.

**Notations.** For the Lebesgue measure on a set $A \subset \mathbb{R}$, the usual $L^2(A)$ inner product and norm are

$$\langle f, g \rangle = \int_A f(x) g(x) dx, \quad \|f\|^2 = \int_A |f(x)|^2 dx.$$

For a set $A \subset \mathbb{R}$, $f_A$ denotes the restriction of $f$ to $A$.

For $m \in \mathbb{N}^*$, $I_m$ is the $m \times m$ identity matrix. For a real $m \times m$ matrix $M \in \mathbb{R}^{m \times m}$, we write $\|M\|_{\text{op}} = \sup_{\|x\|_{2,m}=1} \|Mx\|_{2,m}$, the spectral (operator) norm induced by the Euclidean norm $\|\cdot\|_{2,m}$
on $\mathbb{R}^m$.

If $A, B$ are symmetric matrices of the same size, the notation $A \prec B$ means that $B - A$ is positive semidefinite, i.e.,

$$ \forall x \in \mathbb{R}^m, \quad x^\top A x \le x^\top B x. $$

## 2. Preliminaries on the model

Fix a time horizon $T > 0$ and let $N \in \mathbb{N}$. Throughout, we adopt the convention that $NT$ denotes the integer $\lfloor NT \rfloor$ whenever an integer is required. Let $(W_t^i)_{t \ge 0}$ and $i = 1, \dots, N$ be i.i.d. standard Brownian motions defined on the same filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ with

$$ \mathcal{F}_t := \sigma(W_s^j : 0 \le s \le t, j = 1, \dots, N), \quad t \ge 0. $$

For each $i \in \{1, \dots, N\}$, we consider the strong solution $(X_t^i)_{0 \le t \le T}$ to

$$ dX_t^i = b(X_t^i) dt + \sigma(X_t^i) dW_t^i, \quad X_0^i = x_0. \qquad (2.1) $$

By construction, ${X_t^i}_{i=1}^N$ are i.i.d. processes with a common law given by the unique strong solution to (2.1) driven by a standard Brownian motion.

### 2.1. Assumptions on $X$.

We introduce two standard classes of drift-diffusion functions:

$$ \mathbb{S}_1 := \left\{ (b, \sigma) : b \in C^1(\mathbb{R}), \sigma \in C^2(\mathbb{R}), \exists C > 0 \text{ such that } |b(x)| + |\sigma(x)| \le C(1 + |x|), \forall x \in \mathbb{R} \right\}, $$

$$ \mathbb{S}_2 := \left\{ (b, \sigma) : b \text{ is globally Lipschitz on } \mathbb{R}, \sigma \text{ is Hölder continuous of order } \alpha \in [1/2, 1] \right\}. $$

**Assumption 1.** The pair of coefficients $(b, \sigma)$ belongs to either $\mathbb{S}_1$ or $\mathbb{S}_2$.

Assumption 1 ensures strong existence and pathwise uniqueness for (2.1), adapted to the filtration $(\mathcal{F}_t)_{t \ge 0}$. The pair $(b, \sigma) \in \mathbb{S}_1$ is classical in the diffusion literature, while $(b, \sigma) \in \mathbb{S}_2$ covers, for instance, the square-root diffusion $\sigma(x) = \sqrt{x_+}$ (CIR model). In either case, there exists a constant $K > 0$ such that

$$ b(x)^2 + \sigma(x)^2 \le K(1+x^2), \quad \forall x \in \mathbb{R}. $$

Moreover, the corresponding Markov transition density $p_t(x, y)$ is jointly continuous on $\mathbb{R}_+ \times (l, r) \times (l, r)$ for the state space $(l, r)$; see Rogers and Williams (2000, Chap. V, §7). Since the initial condition is deterministic, it follows that for every integer $k \ge 0$ and all $t \ge 0$

$$ \sup_{0 \le u \le t} \mathbb{E}[(X_u^1)^{2k}] = \sup_{0 \le u \le t} \int y^{2k} p_u(x_0, y) dy < \infty. \qquad (2.2) $$

The next condition will be used to derive the risk bounds later in Section 4.2 and to justify the data-driven model selection rule in Section 4.3.

**Assumption 2.** There exist constants $0 < c_0 \le c_1 < \infty$ such that

$$ c_0 \le \sigma(x) \le c_1 \quad \text{for all } x \in A, $$

where $A$ denotes the estimation domain in Section 3.

Assumption 2 ensures a uniform control of the variance terms appearing in Proposition 4.1.
## 2.2. Observations.

We observe a censored version of $X$ through a censoring process $C$ that satisfies the following condition:

**Assumption 3.** Let $\{(\widetilde{W}_t^i)_{t \ge 0}\}_{i \ge 1}$ be i.i.d. standard Brownian motions defined on $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ and independent of $\{(W_t^i)_{t \ge 0}\}_{i \ge 1}$. For each $i \ge 1$, $C^i$ is the (a.s. unique) strong solution to

$$dC_t^i = b_C(C_t^i) dt + \sigma_C(C_t^i) d\widetilde{W}_t^i,$$

with $C_0^i$ independent of $(W^i, \widetilde{W}^i)$ and $(b_C, \sigma_C)$ belongs to $\mathbb{S}_1$ or to $\mathbb{S}_2$.

We set

$$Y_t^i := X_t^i \wedge C_t^i, \quad \delta_t^i := \mathbb{1}_{\{X_t^i < C_t^i\}}, \quad 0 \le t \le T.$$

Our aim is to estimate the drift $b$ using only the data $(Y^i, \delta^i)_{i=1}^N$.

## 2.3. Probabilistic preliminaries.

Define the (time-dependent) survival function of the censoring process by

$$\bar{G}_t(x) := \mathbb{P}(C_t^1 \ge x), \quad x \in \mathbb{R}, t \in [0, T].$$

We also introduce the averaged survival-weighted function

$$\bar{f}_T(x) = \frac{1}{T} \int_0^T \bar{G}_t(x) p_t(x_0, x) dt, \quad x \in \mathbb{R}. \qquad (2.3)$$

For every $t \in [0, T]$ and $x \in \mathbb{R}$, we have $0 \le \bar{G}_t(x) \le 1$; hence $\bar{G}(x) \in L^1([0, T])$. Consequently, by Fubini's theorem, the integral in (2.3) is well-defined and

$$0 \le \bar{f}_T(x) \le \frac{1}{T} \int_0^T p_t(x_0, x) dt < \infty,$$

combining this with (2.2), we obtain that $\bar{f}_T \in L^1(\mathbb{R})$. Moreover, by analogy with survival analysis, $\bar{f}_T$ may be viewed as a sub-density (since $\int_{\mathbb{R}} \bar{f}_T(x) dx \le 1$). Using Assumption 1 and the upper bound (2.2), for all $k \in \mathbb{N}$,

$$\|b\|_{L^{2k}(\bar{f}_T)}^2 + \|b\|_{L^{2k}(\bar{f}_T)}^2 = \int_{\mathbb{R}} (b^{2k}(y) + \sigma^{2k}(y)) \bar{f}_T(y) dy \le \int_{\mathbb{R}} (b^{2k}(y) + \sigma^{2k}(y)) \frac{1}{T} \int_0^T p_t(x_0, y) dt dy \\ = \frac{1}{T} \mathbb{E} \left[ \int_0^T (b^{2k}(X_t^1) + \sigma^{2k}(X_t^1)) dt \right] < \infty,$$

Thus, the constants in our main results, Proposition 4.1 and Theorems 4.3, are well defined.

Write $Y_t^i = X_t^i - (X_t^i - C_t^i)^+$ with $(x)^+ = \max\{x, 0\}$. By Tanaka's formula (e.g. Revuz and Yor, 2013, Th.VI.1.2), $(X^i - C^i)^+$ is a continuous semimartingale. In particular, all stochastic integrals with respect to $Y^i$ that appear below are well defined. The following lemma plays a crucial role in the subsequent analysis and is proved in Section 7.1.

**Lemma 2.1.** Let $V$ be a continuous $\mathbb{L}^2$-bounded semimartingale and let $t \le T$. Then, for every $i \in \{1, ..., N\}$,

$$\int_0^t V_u \delta_u^i dY_u^i = \int_0^t V_u \delta_u^i dX_u^i.$$
Let $i \in \{1, ..., N\}$ and $L_t^x(Y^i)$ denote the (right-continuous) local time of $Y^i$ at level $x$, and define the uncensored occupation density and measure

$$ \ell_T^i(x) := \int_0^T \delta_t^i dL_t^x(Y^i) \ge 0, \quad \mu_T^i(dy) := \ell_T(y)^i dy. \qquad (2.4) $$

We note $\hat{\mu}_T := \frac{1}{NT} \sum_{i=1}^N \mu_T^i$.

Let $\varphi$ be a bounded Borel function. By the occupation-time identity for continuous semimartingales (see Revuz and Yor, 2013, Th. VI.1.6 and Cor. VI.1.7),

$$ \int_0^T \delta_t^i \varphi(Y_t^i) dt = \int_{\mathbb{R}} \varphi(Y_t^i) \ell_T^i(y) dy = \int_{\mathbb{R}} \varphi(Y_t^i) \mu_T^i(dy), $$

moreover, since $\delta_t^i \varphi(Y_t^i) = \delta_t^i \varphi(X_t^i)$ and $C^i$ are independent of $X^i$,

$$ \mathbb{E}[\delta_t^i \varphi(Y_t^i)] = \mathbb{E}[\delta_t^i \varphi(X_t^i)] = \mathbb{E}[\mathbb{E}[\delta_t^i | X_t^i] \varphi(X_t^i)] = \mathbb{E}[\overline{G}_t(X_t^i) \varphi(X_t^i)], $$

then

$$ \mathbb{E}[\overline{G}_t(X_t^i) \varphi(X_t^i)] = \int_{\mathbb{R}} \varphi(x) \overline{G}_t(x) p_t(x_0, x) dx. $$

Therefore,

$$ \begin{aligned} \mathbb{E}\left[\int_{\mathbb{R}} \varphi(y) \hat{\mu}_T(dy)\right] &= \frac{1}{NT} \sum_{i=1}^N \mathbb{E}\left[\int_0^T \delta_t^i \varphi(Y_t^i) dt\right] = \frac{1}{T} \int_0^T \int_{\mathbb{R}} \varphi(x) \overline{G}_t(x) p_t(x_0, x) dx dt \\ &= \int_{\mathbb{R}} \varphi(x) \overline{f}_T(x) dx, \end{aligned} \qquad (2.5) $$

with $\overline{f}_T$ as in (2.3). In words, $\hat{\mu}_T$ is the empirical (pathwise) version of $\overline{f}_T(x)$ $dx$.

In our continuous-time observation scheme, $\sigma$ is recoverable from $(\langle Y^1 \rangle_t)_{t \ge 0}$, where $(\langle Y^1 \rangle_t)_{t \ge 0}$ is the quadratic variation of $Y^1$. The following Lemma is proved in Section 7.2:

**Lemma 2.2.** Under Assumptions 1 and 3,

(a) If, for some $x \in \mathbb{R}$, one has either $b_C(x) > b(x)$, $\sigma(x) \ne 0$, or $\sigma_C(x) \ne 0$, then, a.s., the set $U := \{t \in [0, T] : \delta_t^1 = 1\}$ has a nonempty interior. In particular, a.s., there exists $I \subset [0, T]$ such that $Y_I^1 = X_I^1$.

(b) By part (a), there exists an open interval $I \subset [0, T]$ on which $X_t^1 < C_t^1$ holds for all $t \in I$. Then, for every $x \in Y^1(I) := \{x \in \mathbb{R} : \exists t \in I, Y_t^1 = x\}$, there exists $t(x) \in I$ with $Y_{t(x)}^1 = x$ such that

$$ \left. \frac{d\langle Y^1 \rangle_t}{dt} \right|_{t=t(x)} = \sigma^2(Y_{t(x)}^1) = \sigma^2(x). \qquad (2.6) $$

(c) Moreover, if $x$ is in the interior of $Y^1(I)$,

$$ \sigma^2(x) = \lim_{\epsilon \downarrow 0} \frac{\int_0^T \delta_t^1 \mathbf{1}_{\{|Y_t^1 - x| \le \epsilon\}} d\langle Y^1 \rangle_t}{\int_0^T \delta_t^1 \mathbf{1}_{\{|Y_t^1 - x| \le \epsilon\}} dt}. \qquad (2.7) $$

*Remark 2.3.* On $I$, for $\epsilon$ small enough, the denominators in (2.7) are strictly positive for $x$ in the interior of $Y^1(I)$ (see the proof of Lemma 2.2).

From now on, we will assume
**Assumption 4.** There exist $x \in \mathbb{R}$, such that $b_C(x) > b(x)$, or $\sigma(x) \neq 0$, or $\sigma_C(x) \neq 0$.

Under Assumption 4, by Lemma 2.2, a.s., there exists $I \subset [0, T]$ such that $Y_{I|I}^1 = X_{I|I}^1$ and $\sigma^2(x)$ is identified pathwise from the quadratic variation of $Y^1$. For clarity, we henceforth treat $\sigma$ as known and focus on the estimation of the drift function $b$. Importantly, the estimator of $b$ does not depend on the exact value of $\sigma$, but only on an upper bound for $\sigma^2$ that enters the penalty term.

### 3. The projection least-squares estimator for censored diffusions

Fix a measurable subset $A \subset \mathbb{R}$ and $m \in \mathbb{N}^+$. Let $\{\varphi_0, \dots, \varphi_{m-1}\}$ be an orthonormal system in $L^2(A)$, and denote by $S_m := \operatorname{span}\{\varphi_0, \dots, \varphi_{m-1}\} \subset L^2(A)$ its $m$-dimensional approximation space. For non-compact $A$, we rely on the classical Laguerre basis (when $A = \mathbb{R}_+$) and the Hermite basis (when $A = \mathbb{R}$). For compact $A$, we will also use the trigonometric (Fourier) basis; see Section 3.1.

We assume that the orthonormal system $\{\varphi_0, \dots, \varphi_{m-1}\}$ satisfies the following condition

**Assumption 5.** The projection space is such that

(1) There exist a constant $c_\varphi > 0$ such that $L(m) \le c_\varphi^2 m, \forall m \ge 1$, where $L(m) = \sup_{x \in A} \sum_{j=0}^{m-1} \varphi_j(x)^2$.

(2) $S_m \subset S_{m'}, \forall m < m'$;

(3) $S_m \subset L^2(\mathbb{R}, \bar{f}_T(x) dx), \forall m \ge 1$

*Remark 3.1.* (i) The linear growth bound in (1) is a convenient (non-sharp) sufficient condition that streamlines risk bounds. It is satisfied in all examples presented in Section 3.1.

(ii) The nesting (2) is automatic when $\{\varphi_j\}_{j \ge 0}$ is fixed, i.e., the $\varphi_j$ does not depend on $m$.

(iii) Condition (3) holds whenever the basis functions are bounded on $A$ since $\bar{f}_T \in L^1(\mathbb{R})$.

Throughout the paper, $L^2(\mathbb{R}, \bar{f}_T(x) dx)$ is endowed with

$$ \langle h, \ell \rangle_{\bar{f}_T} := \int_{\mathbb{R}} h(x) l(x) \bar{f}_T(x) dx, \quad \|h\|_{\bar{f}_T}^2 = \langle h, h \rangle_{\bar{f}_T}, \quad \text{for } h, \ell \in L^2(\mathbb{R}, \bar{f}_T(x) dx). $$

For their empirical counterpart, let $h, \ell \in L^2(\mathbb{R}, \bar{f}_T(x) dx)$ be bounded (to ensure the integrability of the following integrals); define

$$ \langle h, \ell \rangle_N := \frac{1}{NT} \sum_{i=1}^{N} \int_{0}^{T} \delta_{t}^{i} h(Y_{t}^{i}) l(Y_{t}^{i}) dt = \frac{1}{NT} \sum_{i=1}^{N} \int_{\mathbb{R}} h(y) l(y) \mu_{T}^{i}(dy). $$

By (2.5), $\mathbb{E}[\langle \cdot, \cdot \rangle_N] = \langle \cdot, \cdot \rangle_{\bar{f}_T}$.

Our aim is to estimate $b$. For $h \in S_m$, define the (observable under censoring, as all terms depend only on $(Y, \delta)$) projection contrast

$$ \gamma_N(h) := \|h\|_N^2 - \frac{2}{NT} \sum_{i=1}^{N} \int_{0}^{T} h(Y_t^i) \delta_t^i dY_t^i. $$

By Lemma 2.1,

$$ \begin{aligned} \gamma_N(h) &= \frac{1}{NT} \sum_{i=1}^{N} \int_{0}^{T} \delta_t^i h^2(X_t^i) dt - \frac{2}{NT} \sum_{i=1}^{N} \int_{0}^{T} h(X_t^i) \delta_t^i dX_t^i \\ &= \frac{1}{NT} \sum_{i=1}^{N} \int_{0}^{T} \delta_t^i h^2(X_t^i) dt - \frac{2}{NT} \sum_{i=1}^{N} \int_{0}^{T} h(X_t^i) \delta_t^i b(X_t^i) dt - \frac{2}{NT} \sum_{i=1}^{N} \int_{0}^{T} h(X_t^i) \delta_t^i \sigma(X_t^i) dW_t^i. \end{aligned} $$
In the complete-observation case ($\delta_t^i \equiv 1$), this reduces to the usual contrast studied for uncensored trajectories (see, e.g., Comte and Genon-Catalot, 2020a).

Moreover,

$$ \mathbb{E}[\gamma_N(h)] = \|h\|_{\bar{f}_T}^2 - 2\langle h, b \rangle_{\bar{f}_T} = \|h - b\|_{\bar{f}_T}^2 - \|b\|_{\bar{f}_T}^2. $$

Thus, minimizing $\gamma_N$ is, on average, equivalent to finding the best approximation of $b$ in $S_m \subset L^2(\mathbb{R}, \bar{f}_T(x) dx)$.

Define the projection estimator

$$ \hat{b}_m := \arg \min_{h \in S_m} \gamma_N(h). \quad (3.1) $$

Write any $h \in S_m$ as $h(x) = \sum_{j=0}^{m-1} \beta_j^{(m)} \varphi_j(x)$, set $\beta^{(m)} := (\beta_0^{(m)}, \dots, \beta_{m-1}^{(m)})^\top$, and introduce the empirical matrices

$$ \hat{\Psi}_m := (\langle \varphi_j, \varphi_k \rangle_N)_{0 \le j, k \le m-1} \in \mathbb{R}^{m \times m}, \quad \hat{Z}_m := \left( \frac{1}{NT} \sum_{i=1}^N \int_0^T \delta_t^i \varphi_j(Y_t^i) dY_t^i \right)_{0 \le j \le m-1} \in \mathbb{R}^m. \quad (3.2) $$

Notice that $\|h\|_N^2 = (\beta^{(m)})^\top \hat{\Psi}_m \beta^{(m)}$; thus, $\gamma_N(h) = (\beta^{(m)})^\top \hat{\Psi}_m \beta^{(m)} - 2(\beta^{(m)})^\top \hat{Z}_m$, and setting the normal equations $\nabla_{\beta} \gamma_N = 2\hat{\Psi}_m \beta^{(m)} - 2\hat{Z}_m$ to zero yields that whenever $\hat{\Psi}_m$ is invertible,

$$ \hat{\beta}^{(m)} = \hat{\Psi}_m^{-1} \hat{Z}_m, \quad \hat{b}_m(x) = \sum_{j=0}^{m-1} \hat{\beta}_j^{(m)} \varphi_j(x). \quad (3.3) $$

The invertibility of the empirical Gram matrix $\hat{\Psi}_m$ and its expectation $\Psi_m = \mathbb{E}[\hat{\Psi}_m]$ requires careful consideration. In difficult cases, even if these matrices are invertible, they may be poorly conditioned. To address this, we adopt the stability condition from Cohen et al. (2013, 2019):

**Assumption 6.** The integer $m \ge 1$ is assumed to satisfy

$$ L(m) (\|\hat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le \frac{c_T NT}{2 \log(NT)}, \quad m \le NT, \quad (3.4) $$

with $c_T := \frac{3 \log(3/2) - 1}{8T}$.

The constant $c_T$ is determined from the proof of Lemma 7.1. For later risk control, set

$$ \Lambda_m := \left\{ L(m) (\|\hat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le \frac{c_T NT}{\log(NT)} \right\}, \quad (3.5) $$

Under (3.4), (3.5) is true with a probability near 1. We will work with the truncated estimator.

$$ \tilde{b}_m := \hat{b}_m \mathbf{1}_{\Lambda_m}. \quad (3.6) $$

### 3.1. Examples of approximation spaces.

We illustrate the construction with three classical bases that verify Assumption 5.

**Trigonometric (Fourier) basis on a compact interval** ($A = [a, b]$). Define

$$ \varphi_0(x) = (b-a)^{-1/2}, \quad \varphi_{2k-1}(x) = \sqrt{\frac{2}{b-a}} \cos\left(\frac{2\pi k(x-a)}{b-a}\right), \quad \varphi_{2k}(x) = \sqrt{\frac{2}{b-a}} \sin\left(\frac{2\pi k(x-a)}{b-a}\right), \quad k \ge 1. $$

This is an orthonormal basis of $L^2(A)$ satisfying Assumption 5.
**Laguerre basis** ($A = \mathbb{R}_+$). For $j \ge 0$, define

$$L_j(x) = \sum_{k=0}^{j} (-1)^k \binom{j}{k} \frac{x^k}{k!}, \quad \ell_j(x) = \sqrt{2} L_j(2x) e^{-x} \mathbf{1}_{x \ge 0}.$$

The family {$\ell_j : j \ge 0$} is an orthonormal basis of $L^2(\mathbb{R}_+)$ satisfying $\|\ell_j\|_\infty \le \sqrt{2}$ for all $j$ (Abramowitz and Stegun (1968, §22.14.12)). Consequently, Assumption 5 is verified.

**Hermite basis** ($A = \mathbb{R}$). For $j \ge 0$ let

$$H_j(x) = (-1)^j e^{x^2} \frac{d^j}{dx^j} (e^{-x^2}), \quad h_j(x) = c_j H_j(x) e^{-x^2/2}, \quad c_j := (2^j j! \sqrt{\pi})^{-1/2}.$$

Then {$h_j : j \ge 0$} is an orthonormal basis of $L^2(\mathbb{R})$ and classical bounds (Abramowitz and Stegun (1968, §22.14.17); Szegö (1939, p. 242); Indritz (1961)) give $\|h_j\|_\infty \le \Phi_0$ with $\Phi_0 = \pi^{-1/4}$. Hence, Assumption 5 is verified.

**Recursive evaluation.** The Laguerre and Hermite polynomials admit fast three-term recursions:

$$L_0(x) = 1, \quad L_1(x) = 1 - x, \quad (j+1)L_{j+1}(x) = (2j+1-x)L_j(x) - jL_{j-1}(x), \quad j \ge 1.$$

$$H_0(x) = 1, \quad H_1(x) = x, \quad H_{j+1}(x) = 2x H_j(x) - 2j H_{j-1}(x), \quad j \ge 1.$$

These formulas are useful for the numerical generation of basis functions.

### 3.2. Regularity classes used for rates.

We collect the function classes that will be assumed when deriving rates.

* For $A = [a, b]$, $s > 0$, write $s = r + \alpha$ with $r = \lfloor s \rfloor$ and $\alpha \in (0, 1]$. The modulus of smoothness of order $r$ is

$$\omega_r(f, t)_2 := \sup_{|h| \le t} \|\Delta_h^r f\|_{L^2(A)}, \quad t > 0,$$

where $\Delta_h^r f(\cdot) = \sum_{k=0}^r (-1)^{r-k} \binom{r}{k} f(\cdot + kh)$ denotes the $r$-th order forward difference operator. For $R > 0$ we define the Besov ball

$$B_{2,\infty}^s(A, R) := \left\{ f \in L^2(A) : \|f\|_{L^2(A)} + \sup_{t>0} t^{-\alpha} \omega_r(f,t)_2 \le R \right\}.$$

* Let {$\ell_j$}$_{j \ge 0}$ be the Laguerre basis on $\mathbb{R}_+$ (resp. {$h_j$}$_{j \ge 0}$ be the Hermite basis on $\mathbb{R}$). For $s > 0$ and $R > 0$, set

$$W_{\text{Lag}}^s(R) := \left\{ f \in L^2(\mathbb{R}_+) : \sum_{j \ge 0} j^s \langle f, \ell_j \rangle^2 \le R \right\}, \quad W_{\text{Herm}}^s(R) := \left\{ f \in L^2(\mathbb{R}) : \sum_{j \ge 0} j^s \langle f, h_j \rangle^2 \le R \right\}.$$

## 4. Risk bounds and data-driven model selection

All risk bounds in this section are non-asymptotic, with explicit constants, and are valid for every finite $N$ and $T$.
### 4.1. Risk bound for a fixed dimension.

Define the $\sigma^2$-weighted Gram matrix

$$ \Psi_{m,\sigma^2} := (\langle \sigma\varphi_j, \sigma\varphi_k \rangle_{\bar{f}_T})_{0\le j,k<m} \in \mathbb{R}^{m \times m}. \quad (4.1) $$

**Proposition 4.1.** Under Assumptions 1, 3-6, the truncated estimator (3.6) satisfies

$$ \mathbb{E}[\|\tilde{b}_m - b_A\|_N^2] \le \inf_{\substack{h \in S_m \\ \text{bias}^2}} \|h - b_A\|_{\bar{f}_T}^2 + \frac{2}{NT} \text{Tr}(\Psi_m^{-1} \Psi_{m,\sigma^2}) + \frac{C_1(T)}{NT}, \quad (4.2) $$

where $C_1(T)$ depends only on $T$, $\|b_A\|_{L^4(\bar{f}_T)}$ and $\|\sigma_A\|_{L^4(\bar{f}_T)}$.

The proof of Proposition 4.1 is given in Section 7.3. The next result provides a control for the trace term. Its proof follows the same arguments as Comte and Genon-Catalot (2021, Prop. 2.2) and is thus omitted for brevity.

**Proposition 4.2.** Under Assumption 2, $\text{Tr}(\Psi_m^{-1}\Psi_{m,\sigma^2}) \le c_1^2 m$.

### 4.2. Rates of convergence.

**Case A** = $[a, b]$. If, for some $s > 0$ and $R > 0$, we assume $b_A \in \mathcal{B}_{2,\infty}^s(A, R)$. Since A is compact, the diffusion transition density $p_t(x_0, \cdot)$ is bounded from above and below, as it is continuous and positive on A for $t \in [0, T]$; the censoring survival $\bar{G}_t(\cdot)$ is uniformly bounded below on A and is trivially bounded from above by 1. Then $\bar{f}_T$ inherits two-sided bounds:

$$ 0 < f_{-} \le \bar{f}_{T}(x) \le f_{+} < \infty \quad \text{for all } x \in A $$

Then standard Fourier approximation results yield

$$ \inf_{h \in S_m} \|h - b_A\|_{\bar{f}_T}^2 \lesssim \inf_{h \in S_m} \|h - b_A\|_{L^2(A)}^2 \lesssim R m^{-2s}. \quad (4.3) $$

Combining Proposition 4.1, Proposition 4.2, and $L(m) \lesssim m$ gives, for all $m$ satisfying (3.4),

$$ \mathbb{E}[\|\tilde{b}_m - b_A\|_N^2] \lesssim R m^{-2s} + \frac{m}{NT} + \frac{1}{NT}. $$

Balancing $m^{-2s}$ and $m/(NT)$ yields the oracle dimension $m^* \asymp (NT)^{1/(2s+1)}$. If $m^*$ also satisfies the stability condition (3.4), then

$$ \mathbb{E}[\|\tilde{b}_{m^*} - b_A\|_N^2] \lesssim R (NT)^{-\frac{2s}{2s+1}}. \quad (4.4) $$

Notice that $NT^{2s}/\log(NT) \to \infty$ as $NT \to \infty$ for any $s > 0$, thus (3.4) is asymptotically satisfied for large $NT$.

**Case A non compact.** For $A = \mathbb{R}_+$ (Laguerre) or $A = \mathbb{R}$ (Hermite), assume that for some $s > 0$ and $R > 0$

$$ b_A \in \mathcal{W}_{\text{Lag}}^s(R) \quad (\text{on } \mathbb{R}_+) \quad \text{or} \quad b_A \in \mathcal{W}_{\text{Herm}}^s(R) \quad (\text{on } \mathbb{R}). $$

This implies the approximation bound $\inf_{h \in S_m} \|h - b_A\|_{\bar{f}_T}^2 \lesssim R m^{-s}$. Together with Proposition 4.2 this yields, for all $m$ satisfying (3.4),

$$ \mathbb{E}[\|\tilde{b}_m - b_A\|_N^2] \lesssim R m^{-s} + \frac{m}{NT} + \frac{1}{NT}. $$

Balancing gives the oracle choice $m^* \asymp (NT)^{1/(s+1)}$. If $m^*$ also satisfies (3.4), then

$$ \mathbb{E}[\|\tilde{b}_{m^*} - b_A\|_N^2] \lesssim R (NT)^{-\frac{s}{s+1}}. \quad (4.5) $$
### 4.3. Data-driven choice of the dimension m.

In the following, we specify the data-driven model selection rule, similar to that of Comte and Genon-Catalot (2020b). A key novelty with respect to these references is that the collections of models $\widehat{\mathcal{M}}_N$ and $\mathcal{M}_N$ involve $\|\Psi_m^{-1}\|_{\text{op}}$ only to the first power instead of being squared.

Introduce the random set

$$ \widehat{\mathcal{M}}_N := \{1 \le m \le NT : c_{\varphi}^2 m (\|\widehat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le c_T \frac{NT}{\log(NT)}\} $$

and its deterministic analog

$$ \mathcal{M}_N := \{1 \le m \le NT : c_{\varphi}^2 m (\|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le \frac{c_T}{2} \frac{NT}{\log(NT)}\}. $$

Let $\kappa > 0$ be a tuning constant whose value will be shown later, and define the penalty

$$ \text{pen}(m) = \kappa \frac{c_1^2 m}{NT}, $$

where $c_1$ is from Assumption 2. By Proposition 4.2, $\text{pen}(m)$ approximates the variance term in Proposition 4.1.

For $h \in S_m$, write $h(x) = \sum_{j=0}^{m-1} \beta_j^{(m)} \varphi_j(x)$ and set $\beta = (\beta_0^{(m)}, \dots, \beta_{m-1}^{(m)})^{\top}$. If $m \in \mathcal{M}_N$, then $\widehat{\Psi}_m$ is invertible, and by (3.3)

$$ \gamma_N(\hat{b}_m) = \min_{h \in S_m} \gamma_N(h) = -Z_m^{\top} \widehat{\Psi}_m^{-1} Z_m = -\hat{\beta}^{\top} \widehat{\Psi}_m \hat{\beta} = -\|\hat{b}_m\|_N^2. \quad (4.6) $$

Therefore, we define the following dimension-selection criterion:

$$ \hat{m} := \arg \min_{m \in \widehat{\mathcal{M}}_N} \left\{ -\|\hat{b}_m\|_N^2 + \text{pen}(m) \right\}, \quad (4.7) $$

and the final estimate is $\hat{b}_{\hat{m}}$, which equals the truncated version (3.6) by construction.

**Theorem 4.3.** Under Assumptions 1-5, there exists $\kappa_0 \ge 0$ such that, if $\kappa \ge \kappa_0$, then

$$ \mathbb{E}[\|\hat{b}_{\hat{m}} - b_A\|_N^2] \le \frac{8}{3} \inf_{m \in \mathcal{M}_N} \left\{ \inf_{h \in S_m} \|h - b_A\|_{\hat{f}_T}^2 + \text{pen}(m) \right\} + \frac{C_2(T)}{NT}, $$

where $C_2(T)$ depends only on $\|b_A\|_{L^4(\bar{f}_T)}$, $\|\sigma_A\|_{L^4(\bar{f}_T)}$ and $T$.

**Remark 4.4.** (1) The numerical value of $\kappa_0$ is an artifact of the proof and is typically conservative (too large); in applications, one may calibrate $\kappa$ through a short pilot experiment, as suggested by Birgé and Massart (2007).

(2) When the upper-bound in Assumption 5 is not satisfied, replace the previous penalty with

$$ \widehat{\text{pen}}_{(\sigma)}(m) := \kappa \frac{m(1 + \|\widehat{\Psi}_m^{-1/2} \widehat{\Psi}_{m,\sigma^2} \widehat{\Psi}_m^{-1/2}\|_{\text{op}})}{NT}, \quad \widehat{\Psi}_{m,\sigma^2} := (\langle \sigma \varphi_j, \sigma \varphi_k \rangle_N)_{0 \le j,k < m}, $$

which leads to an analog of Theorem 4.3; see Comte and Genon-Catalot (2020a, §3.2).

If $A = [a, b]$, $b_A \in \mathcal{B}_{2,\infty}^s(A, R)$ with $s > 0$, then combining Theorem 4.3 with (4.3) yields the non-asymptotic oracle inequality.

$$ \mathbb{E}[\|\hat{b}_{\hat{m}} - b_A\|_N^2] \lesssim \inf_{m \in \mathcal{M}_N} \left\{ R m^{-2s} + \frac{m}{NT} \right\} + \frac{1}{NT}. $$
Let $m^* \asymp (NT)^{1/(2s+1)}$ denote the balancing choice. Since $NT^{\frac{2s}{2s+1}}/\log(NT) \to \infty$ as $NT \to \infty$ for any $s > 0$, then $m^* \in \mathcal{M}_N$ for large $NT$, and the rate $R(NT)^{-2s/(2s+1)}$ follows.

If A is non-compact and $b_A \in W_{\text{Lag}}^s(R)$ or $b_A \in W_{\text{Herm}}^s(R)$ for some $s > 0$, then, under Assumption 5, Theorem 4.3 together with the bias bound $O(Rm^{-s})$ gives

$$ \mathbb{E}[\|\hat{b}_{\hat{m}} - b_A\|_N^2] \lesssim \inf_{m \in \mathcal{M}_N} \left\{ Rm^{-s} + \frac{m}{NT} \right\} + \frac{1}{NT}. $$

Let $m^* \asymp (NT)^{1/(s+1)}$. If $m^* \in \mathcal{M}_N$, then the rate $R(NT)^{-s/(s+1)}$ follows.

## 5. Simulation study

We assess the finite-sample performance of the proposed censored projection least-squares estimator across three benchmark diffusions and three censoring schemes. Unless stated otherwise, the condition of (2.1) is fixed to $x_0 = 0$. Accuracy is summarized using model-specific mean integrated squared error (MISE) tables computed with $K = 100$ Monte Carlo replications.

### 5.1. Models.

We consider one-dimensional (2.1) and estimate $b$ using the model-selection rule in (4.7) over bases adapted to the state space:

* **Ornstein-Uhlenbeck (OU):** $b(x) = -x$, $\sigma(x) \equiv 1$; estimation on a compact $A$ with the trigonometric (Fourier) basis.

* **Double-well:** $b(x) = -(x^3 - x)$, $\sigma(x) \equiv 0.7$; estimation on $\mathbb{R}$ with the Hermite basis.

* **Cox-Ingersoll-Ross (CIR):** $b(x) = 2(0.5 - x)$, $\sigma(x) = \sqrt{x_+}$; estimation on $\mathbb{R}_+$ with the Laguerre basis.

OU satisfies Assumption 1 and, on compact $A$, Assumption 2; the Fourier basis verifies Assumption 5. For the double-well model, Assumptions 5, 3-4 hold, and $\sigma$ is bounded, but $b$ is not globally Lipschitz; the drift is dissipative, ensuring a unique nonexplosive solution with finite moments, which suffices for our simulations. For CIR, Assumption 1 holds, and the Laguerre basis satisfies Assumption 5; however, $\sigma$ is unbounded on $A = \mathbb{R}_+$, so the selection uses the modified penalty $\widehat{\text{pen}}_{(\sigma)}$ from the remark in Section 4.3.

We observe $Y_t^i = X_t^i \wedge C_t^i$, $\delta_t^i = \mathbf{1}_{\{X_t^i < C_t^i\}}$, $t \in [0, T]$, $i \in \{1, ..., N\}$, with three censoring processes; in each case, one parameter is left free to be calibrated later (Section 5.2):

* **Constant threshold:** $C_t \equiv c_0$ (free parameter: $c_0$).

* **OU threshold (oucensor):** $dC_t = (1 - C_t)dt + \tau dB_t$, with $C_0 = 1$ fixed and a free parameter: $\tau$.

* **Drifted Brownian motion (driftedbm):** $dC_t = \mu dt + \tau dB_t$, with $\mu = 0$, $C_0 = 1$ fixed and free parameters: $\tau$.

Illustrations of estimators and sample paths under these mechanisms are shown in Figure 5.3.

### 5.2. Methods, implementation, and metrics.

We target a common uncensored fraction $\bar{\delta}_T \approx 0.8$ across all (model, censor) combinations. Define

$$ \bar{\delta}_T = \frac{1}{T} \int_0^T \mathbb{P}(X_t < C_t) dt \quad \text{and its estimator} \quad \hat{\bar{\delta}}_T = \frac{1}{NT} \sum_{i=1}^N \int_0^T \delta_t^i dt, $$
with $\hat{\rho}_T = 1 - \hat{\delta}_T$. We tune the free parameter of each censor (i.e., c or $\tau$) by pilot simulations so that $\hat{\delta}_T \approx 0.8$. Here, we take $N = 50, T = 50$ with $K = 100$ repetitions. Table 5.1 reports the tuned parameter values obtained by linear interpolation of the rates in Table D.2.

<table><caption>TABLE 5.1. Calibrated censor parameters achieving $\hat{\delta}_T \approx 0.8$ (see Table D.2 for underlying rate curves).</caption><thead><tr><th>Model</th><th>Constant</th><th>OU-censor</th><th>DriftedBM</th></tr></thead><tbody><tr><td>OU</td><td>c<sub>0</sub> = 0.58</td><td>τ = 1.45</td><td>τ = 0.68</td></tr><tr><td>Double-well</td><td>c<sub>0</sub> = 0.86</td><td>τ = 1.10</td><td>τ = 0.56</td></tr><tr><td>CIR</td><td>c<sub>0</sub> = 0.77</td><td>τ = 0.86</td><td>τ = 0.40</td></tr></tbody></table>

We implement the censored pLS estimator with data-driven dimension selection (4.7). For bounded $\sigma$ (OU, Double-well), we use the penalty pen($m$) = $\kappa c_1^2 m / (NT)$ with

$$ \kappa = 1 \text{ (trigonometric basis), } \quad \kappa = 2 \text{ (Hermite basis).} $$

For CIR (unbounded $\sigma$ on $\mathbb{R}_+$), we use the modified penalty $\widehat{\text{pen}}_{(\sigma)}$ (Section 4.3), with

$$ \kappa = 2 \text{ (Laguerre basis).} $$

The censored (path- and time-averaged) MISE is

$$ \text{MISE} = \frac{\sum_{i=1}^{N} \int_{0}^{T} \delta_{t}^{i} (\hat{b}(X_{t}^{i}) - b(X_{t}^{i}))^{2} dt}{\sum_{i=1}^{N} \int_{0}^{T} \delta_{t}^{i} dt} = \frac{\int (\hat{b} - b)^{2}(x) \hat{\mu}_{T}(dx)}{\hat{\mu}_{T}(\mathbb{R})}, $$

where $\hat{\mu}_T$ is the empirical occupation measure; by (2.5), its numerator concentrates around $\|\hat{b}-b\|_{\hat{J}_T}^2$ up to the scale factor $\hat{\delta}_T$. We vary the information budget $NT \in \{10, 100, 500, 2000, 5000, 10000\}$. For each $NT$, we choose $(N, T)$ on a regular grid with $N \approx \sqrt{NT}$ and $T = NT/N$; all time integrals in the definition of the estimator and of the MISE are discretized using Riemann sums with a time step $\Delta t = 0.01$. We report MISE over $K = 100$ replications; see Tables D.3–D.5, and we show estimator overlays from $k = 25$ replications in Figure 5.3. Log-log MISE plots for OU and Double-well are in Figures 5.1–5.2.

### 5.3. Results.

#### Qualitative findings (across models and censoring modes).

* **Risk vs. information size.** Uncensored MISE decreases steadily with $NT$ for all models (Tables D.3–D.5). The log-log plots for OU and Double-well (Figures 5.1–5.2) show near-linear decay for some censoring modes, indicating stable convergence rates for the pLS estimator.

* **Effect of the censoring mechanism.** At a fixed censoring level, driftedbm yields the lowest MISE, oucensor is intermediate, and the constant threshold performs worst; see Tables D.3–D.5 and overlays in Figure 5.3. This superior performance can be attributed to the higher variability and mobility of the drifted-BM threshold, which produces more diverse uncensored regions along the trajectory, thereby improving the effective coverage of the state space and reducing estimation bias.

* **Bias from hard clipping.** With a constant threshold, the risk plateaus as $NT$ grows (notably in the OU and CIR tables), which is consistent with design bias resulting from limiting the region above the threshold. In contrast, stochastic thresholds create moving uncensored windows that better cover the support of $X$, thereby reducing bias (Figure 5.3).
* **Model dependence.** The qualitative ranking of censoring modes is stable across compact (OU/Fourier) and non-compact (Double-well/Hermite, CIR/Laguerre) settings. The gap is most pronounced in non-compact domains, where clipping removes mass in high-$x$ regions (Tables D.4-D.5).

**Figures and tables.** Figure 5.3 shows the estimator overlays (green) versus the true drift (red) across censoring mechanisms, along with representative trajectories when available. Tables D.3-D.5 report MISE versus *NT* for OU, Double-well, and CIR. Figure 5.1 (OU/Fourier) and Figure 5.2 (Double-well/Hermite) display log-log MISE trends; CIR behaves similarly to Double-well and is omitted for brevity.

<figure><img src="image_3.png" /><figcaption>FIGURE 5.1. OU model (Fourier basis): MISE vs. *NT*. Log-log plot of MISE against the information size *NT*, comparing censoring modes (constant, OU-censor, drifted BM) to the uncensored baseline. Slopes (rates) are reported in the legend.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIGURE 5.2. Double-well model (Hermite basis): MISE vs. *NT*. Log-log plot of MISE against the information size *NT*, comparing censoring modes (constant, OU-censor, drifted BM) to the uncensored baseline. Slopes (rates) are reported in the legend.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>(A) OU with constant threshold: estimator overlays.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(C) OU with drifted-BM threshold: estimator overlays.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(E) Double-well with OU-censor: estimator overlays.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(G) CIR with drifted-BM threshold: estimator overlays.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(B) OU with constant threshold: one trajectory.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(D) OU with drifted-BM threshold: one trajectory.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(F) Double-well with OU-censor: one trajectory.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(H) CIR with drifted-BM threshold: one trajectory.</figcaption></figure>

<caption>FIGURE 5.3. Estimator overlays (green) vs. true drift (red) under different censoring mechanisms, and illustrative trajectories. Shadowed regions in some panels indicate areas that are strongly affected by censoring.</caption>
## 6. Concluding remarks

(1) Section 3 treats $\sigma$ as known; however, in practice, the function $\sigma^2$ is only identifiable on random intervals determined by the observed trajectories. Nevertheless, one can construct an estimator $\hat{\sigma}$ and plug it into the penalty. Starting from Lemma 2.2, a natural many-path analog of (2.7) is to average the local ratio across i.i.d. paths. For $\varepsilon > 0$ and $x \in (\text{supp } \bar{f}_T)^\circ$, define

$$ \hat{\sigma}_{N, \varepsilon}^2(x) := \frac{\sum_{i=1}^{N} \int_{0}^{T} \delta_t^i \mathbf{1}_{\{|Y_t^i - x| \le \varepsilon\}} d\langle Y^i \rangle_t}{\sum_{i=1}^{N} \int_{0}^{T} \delta_t^i \mathbf{1}_{\{|Y_t^i - x| \le \varepsilon\}} dt} $$

By the law of large numbers, together with (2.5), as $N \to \infty$ and $\varepsilon \downarrow 0$,

$$ \hat{\sigma}_{N, \varepsilon}^2(x) \xrightarrow[N \to \infty]{a.s.} \frac{\int_{|y-x| \le \varepsilon} \sigma^2(y) \bar{f}_T(y) dy}{\int_{|y-x| \le \varepsilon} \bar{f}_T(y) dy} \xrightarrow{\varepsilon \downarrow 0} \sigma^2(x), \quad (6.1) $$

for every $x \in (\text{supp } \bar{f}_T)^\circ$. This simple construction suggests that a consistent plug-in estimation of the diffusion coefficient is feasible. This approach warrants further theoretical and empirical investigation, both for its theoretical properties and for its potential to improve the practical implementation of penalized least-squares estimators.

(2) In many practical settings, only a single long trajectory is observed rather than multiple independent copies. When only discrete-time samples are available, the analysis requires combining the ergodic properties of the process with the effects of discretization. The main challenge is to control the concentration of empirical functionals based on $\{\delta_{t_k} \varphi(Y_{t_k})\}$ under mixing assumptions while quantifying the additional error introduced by the discretization step. Developing oracle-type inequalities in this ergodic framework and clarifying the role of censoring in the effective observation time remain open directions. In this setting, the diffusion coefficient $\sigma$ can no longer be treated as known, and its estimation becomes necessary as part of the procedure.

(3) Throughout, we have assumed that the censoring mechanism is independent of the underlying diffusion process. In more realistic scenarios, censoring may depend on the state, leading to a design distorted by the conditional survival function $\bar{G}_t(x)$. An appealing strategy is to introduce inverse-probability weighting with an estimated $\bar{G}_t$, but this raises several questions: how can one ensure the identifiability of $\sigma$ when censoring depends strongly on $X$? what positivity conditions are needed, and how do estimation errors in $\bar{G}_t$ interact with the drift estimation step. Providing a rigorous treatment of these issues is a natural and practically relevant line of future work.

## Acknowledgements

I am deeply grateful to my PhD advisors, Fabienne Comte and Nicolas Marie, for their guidance, insightful discussions, and constant encouragement throughout the development of this work.
## 7. Proofs

### 7.1. *Proof of Lemma 2.1.*

Fix $i \in \{1, \dots, N\}$ and set $U_t^i := X_t^i - C_t^i$. Observe that

$$Y_t^i = X_t^i \wedge C_t^i = \frac{1}{2}(X_t^i + C_t^i - |X_t^i - C_t^i|), \quad t \in [0, T]. \quad (7.1)$$

By Tanaka's formula (see Revuz and Yor (2013, Th. VI.1.2)),

$$|U_t^i| = |U_0^i| + \int_0^t \operatorname{sgn}(U_u^i) dU_u^i + L_t^0(U^i), \quad (7.2)$$

where $\operatorname{sgn}(u) = \mathbf{1}_{\{u>0\}} - \mathbf{1}_{\{u<0\}}$ and $L^0(U^i)$ are the local times of $U^i$ at 0. Combining (7.1) and (7.2) yields

$$\begin{aligned} Y_t^i &= \frac{1}{2} (X_t^i + C_t^i - |U_0^i| - \int_0^t \operatorname{sgn}(U_u^i) dU_u^i - L_t^0(U^i)) \\ &= -\frac{1}{2} |X_0^i - C_0^i| - \frac{1}{2} L_t^0(X^i - C^i) + \frac{1}{2} \int_0^t [1 - \operatorname{sgn}(X_u^i - C_u^i)] dX_u^i + \frac{1}{2} \int_0^t [1 + \operatorname{sgn}(X_u^i - C_u^i)] dC_u^i. \end{aligned} \quad (7.3)$$

Let $V$ be a continuous semimartingale bounded in $L^2$. Multiplying (7.3) by $V_u \delta_u^i$ and integrating, we obtain

$$\begin{aligned} \int_0^t V_u \delta_u^i dY_u^i &= -\frac{1}{2} \int_0^t V_u \delta_u^i dL_u^0(X^i - C^i) \\ &\quad + \frac{1}{2} \int_0^t V_u \delta_u^i [1 - \operatorname{sgn}(X_u^i - C_u^i)] dX_u^i + \frac{1}{2} \int_0^t V_u \delta_u^i [1 + \operatorname{sgn}(X_u^i - C_u^i)] dC_u^i. \end{aligned} \quad (7.4)$$

By the support property of local time (see Revuz and Yor (2013, Cor. VI.1.6–VI.1.7)), the Stieltjes measure $dL_u^0(X^i - C^i)$ is supported on $\{u : X_u^i = C_u^i\}$. On this set, $\delta_u^i = \mathbf{1}_{\{X_u^i < C_u^i\}}$ is zero; hence

$$\int_0^t V_u \delta_u^i dL_u^0(X^i - C^i) = 0. \quad (7.5)$$

On $\{\delta_u^i = 1\}$, we have $X_u^i < C_u^i$; thus, $\operatorname{sgn}(X_u^i - C_u^i) = -1$ and

$$\delta_u^i [1 - \operatorname{sgn}(X_u^i - C_u^i)] = 2\delta_u^i, \quad \delta_u^i [1 + \operatorname{sgn}(X_u^i - C_u^i)] = 0.$$

Plugging these and (7.5) into (7.4) gives

$$\int_0^t V_u \delta_u^i dY_u^i = \int_0^t V_u \delta_u^i dX_u^i. \quad (7.6)$$

Since $i$ and $t$ were arbitrary, (7.6) proves the claim.

### 7.2. *Proof of Lemma 2.2.*

Write

$$\begin{aligned} dY_t^1 &= \frac{1}{2} \left( [1 - \operatorname{sgn}(X_t^1 - C_t^1)] dX_t + [1 + \operatorname{sgn}(X_t^1 - C_t^1)] dC_t \right) - \frac{1}{2} dL_t^0(X^1 - C^1) \\ &= \frac{1}{2} \left( 2\delta_t^1 dX_t^1 + 2(1 - \delta_t^1) dC_t \right) - \frac{1}{2} dL_t^0(X^1 - C^1) \\ &= \delta_t^1 dX_t^1 + (1 - \delta_t^1) dC_t^1 - \frac{1}{2} dL_t^0(X^1 - C^1), \end{aligned} \quad (7.7)$$
Let $C^1 := C_0 + M^C + A^C$ be the canonical decomposition of $C^1$. Then, the local martingale part of $Y^1$ is

$$dM_t^Y = \delta_t^1 \sigma(X_t^1) dW_t^1 + (1 - \delta_t^1) dM_t^C.$$

Independence of $X$ and $C$ implies that $\langle W, M^C \rangle_t \equiv 0$, and then

$$d\langle Y^1 \rangle_t = \delta_t^1 \sigma^2(X_t^1) dt + (1 - \delta_t^1) d\langle C^1 \rangle_t. \quad (7.8)$$

In particular, on $\{\delta_t^1 = 1\}$, we have $Y_t^1 = X_t^1$, and thus

$$d\langle Y^1 \rangle_t = \sigma^2(Y_t^1) dt \quad \text{on } \{\delta_t^1 = 1\}. \quad (7.9)$$

To prove (a), consider $F := \{(c, x) \in \mathbb{R}^2 : \langle (c, x), (-1, 1) \rangle \ge 0\} = \{(c, x) \in \mathbb{R}^2 : c \le x\}$. The stochastic invariance criterion (see Milian (1995, Th. 1); cf. Aubin and Da Prato (1990); Coutin and Marie (2017)) states that for every $C_0^1 \le X_0^1$ and $t \in [0, T]$, $(C_t^1, X_t^1) \in F$ if and only if, on the boundary $\{c = x\}$, one has $b_C(c) - b(x) \le 0$ and $\sigma_C(c)^2 + \sigma(x)^2 = 0$. If there exists $x$ such that $b_C(x) > b(x)$, or $\sigma(x) \ne 0$, or $\sigma_C(x) \ne 0$, the condition fails at $(x, x)$; hence, $(C_t^1, X_t^1)$ exits $F$ at some $t_0 > 0$, i.e., $X_{t_0}^1 < C_{t_0}^1$. By continuity, there is a nonempty open interval $I \subset [0, T]$ on which $X_t^1 < C_t^1$ holds for all $t \in I$, so $\delta_t^1 = 1$, and thus $Y_t^1 = X_t^1$ on $I$ a.s. This proves (a).

To prove (b), fix $x \in Y^1(I) := \{Y_t^1 : t \in I\}$. There exists $t(x) \in I$ with $Y_{t(x)}^1 = x$. By the continuity of $Y^1$ and $\sigma^2$, the map $t \mapsto \sigma^2(Y_t^1)$ is continuous at $t(x)$. Evaluating the density of $d\langle Y^1 \rangle_t$ with respect to $dt$ at $t(x)$ in (7.9) yields

$$\left. \frac{d\langle Y^1 \rangle_t}{dt} \right|_{t=t(x)} = \sigma^2(Y_{t(x)}^1) = \sigma^2(x),$$

which is (2.6).

For (c), fix $x$ in the interior of $Y^1(I)$. Then there exists $t(x) \in I$ with $Y_{t(x)}^1 = x$, and by the continuity of $Y^1$, there is $\Delta > 0$ such that, for all sufficiently small $\varepsilon > 0$,

$$|Y_t^1 - x| \le \varepsilon \quad \text{for all } t \in [t(x) - \Delta, t(x) + \Delta] \subset I.$$

In particular,

$$\int_0^T \mathbf{1}_{\{|Y_t^1 - x| \le \varepsilon\}} dt \ge 2\Delta > 0$$

for all $\varepsilon$ small enough. Let $\ell_T(\cdot)$ denote the (uncensored) occupation density of $Y^1$ on $I$ and recall the occupation-time formula (see Revuz and Yor (2013, Th. VI.1.6 and Cor. VI.1.7)): for any bounded Borel $g$,

$$\int_0^T \delta_t^1 g(Y_t^1) dt = \int_{\mathbb{R}} g(y) \ell_T(y) dy. \quad (7.10)$$

Applying (7.10) with $g = \mathbf{1}_{\{|y-x| \le \varepsilon\}}$ and $g = \sigma^2(y)\mathbf{1}_{\{|y-x| \le \varepsilon\}}$, and using (7.9), we obtain, for every $\varepsilon > 0$,

$$\frac{\int_0^T \delta_t^1 \mathbf{1}_{\{|Y_t^1 - x| \le \varepsilon\}} d\langle Y^1 \rangle_t}{\int_0^T \delta_t^1 \mathbf{1}_{\{|Y_t^1 - x| \le \varepsilon\}} dt} = \frac{\int_{\{|y-x| \le \varepsilon\}} \sigma^2(y) \ell_T(y) dy}{\int_{\{|y-x| \le \varepsilon\}} \ell_T(y) dy}. \quad (7.11)$$

Since $x \in \text{int} Y^1(I)$, the denominator in (7.11) is strictly positive for all small $\varepsilon$. The measure $\ell_T(y) dy$ is absolutely continuous with respect to Lebesgue measure, so the Lebesgue differentiation
theorem (see, e.g., Stein and Shakarchi (2009, Ch. 1-2)) applied to $f(y) = \sigma^2(y)$ and $\nu(dy) = \ell_T(y) dy$ gives

$$ \lim_{\epsilon \downarrow 0} \frac{\int_{\{|y-x| \le \epsilon\}} \sigma^2(y) \ell_T(y) dy}{\int_{\{|y-x| \le \epsilon\}} \ell_T(y) dy} = \sigma^2(x), $$

which proves (2.7).

### 7.3. Proof of Proposition 4.1.

Throughout this proof, the dimension $m$ satisfies the stability condition (3.4). In particular, $\Psi_m$ is invertible, and

$$ \Omega_m := \left\{ \left| \frac{\|h\|_N^2}{\|h\|_{\hat{f}_T}^2} - 1 \right| \le \frac{1}{2} \text{ for all } h \in S_m \right\} = \left\{ \|\Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} - \text{Id}_m\|_{\text{op}} \le \frac{1}{2} \right\} \quad (7.12) $$

(see Lemma A.1).

**Risk decomposition.** Set

$$ T_1 := \|\hat{b}_m - b_A\|_N^2 \mathbf{1}_{\Lambda_m \cap \Omega_m}, \quad T_2 := \|\hat{b}_m - b_A\|_N^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}, \quad T_3 := \|b_A\|_N^2 \mathbf{1}_{\Lambda_m^c}. $$

We bound $\mathbb{E}[T_1] + \mathbb{E}[T_2] + \mathbb{E}[T_3]$.

Main term $T_1$. Let $b_{A,m} := \Pi_m^{(N)}(b_A)$ denote the orthogonal projection of $b_A$ onto $S_m$ with respect to the empirical inner product $\langle \cdot, \cdot \rangle_N$. By the Pythagorean identity associated with this projection, we can decompose the empirical risk as

$$ \|\hat{b}_m - b_A\|_N^2 = \underbrace{\|\hat{b}_m - b_{A,m}\|_N^2}_{=:M_1} + \underbrace{\|b_{A,m} - b_A\|_N^2}_{=:M_2}. $$

Taking expectations and recalling that for any functions $h$ and $\mathbb{E}[\|h\|_N^2] = \|h\|_{\hat{f}_T}^2$, we obtain

$$ \mathbb{E}[M_2] = \|b_{A,m} - b_A\|_{\hat{f}_T}^2 \le \inf_{h \in S_m} \|h - b_A\|_{\hat{f}_T}^2. $$

On the event $\Lambda_m$, the matrix $\hat{\Psi}_m$ is invertible, so that

$$ \hat{b}_m(x) = \sum_{j=0}^{m-1} \hat{\beta}_j^{(m)} \varphi_j(x), $$

where the coefficients $\hat{\beta}^{(m)}$ are given by (3.3). For $b_{A,m} := \Pi_m^{(N)}(b_A)$, write $b_{A,m}(x) = \sum_{j=0}^{m-1} \hat{a}_j^{(m)} \varphi_j(x)$ with $a^{(m)} = (a_0^{(m)}, \dots, a_{m-1}^{(m)})^\top$. By definition, $b_{A,m}$ is the minimizer of $\|h - b_A\|_N^2$ over $h \in S_m$, so $(\hat{a}_j^{(m)})_{0 \le j \le m-1}$ solve the normal equations

$$ \frac{\partial}{\partial a_k^{(m)}} \left\| \sum_{j=0}^{m-1} a_j^{(m)} \varphi_j - b_A \right\|_N^2 = 0, \quad k = 0, \dots, m-1, $$

that is,

$$ \sum_{j=0}^{m-1} \hat{a}_j^{(m)} \langle \varphi_j, \varphi_k \rangle_N = \langle b_A, \varphi_k \rangle_N, \quad k = 0, \dots, m-1. $$

In matrix form, $\hat{\Psi}_m a^{(m)} = (\langle \varphi_k, b_A \rangle_N)_{k<m}$. On $\Lambda_m$, $\hat{\Psi}_m$ is invertible; hence

$$ \hat{a}^{(m)} = \hat{\Psi}_m^{-1} (\langle \varphi_j, b_A \rangle_N)_{j<m} \quad \text{and} \quad b_{A,m}(x) = \sum_{j=0}^{m-1} \hat{a}_j^{(m)} \varphi_j(x). $$
Subtracting these two expressions gives

$$ \hat{\beta}^{(m)} - \hat{a}^{(m)} = \hat{\Psi}_{m}^{-1} E_{m}, $$

where

$$ (E_m)_j = \frac{1}{NT} \sum_{i=1}^{N} \int_{0}^{T} \delta_s^i \varphi_j(X_s^i) \sigma(X_s^i) dW_s^i. $$

For $h = \sum_{j=0}^{m-1} c_j \varphi_j$, one has $\|h\|_N^2 = (c^{(m)})^\top \hat{\Psi}_m c^{(m)}$, thus

$$ M_1 = E_m^\top \hat{\Psi}_m^{-1} E_m = (\Psi_m^{1/2} E_m)^\top M^{-1} (\Psi_m^{1/2} E_m), \quad M := \Psi_m^{1/2} \hat{\Psi}_m^{-1} \Psi_m^{1/2}. $$

On $\Omega_m$, by Lemma A.1, all eigenvalues of $M^{-1}$ lie in $[2/3, 2]$, whence $M_1 \le 2 E_m^\top \hat{\Psi}_m^{-1} E_m$. Using independence across $i$ and Itô isometry,

$$ \mathbb{E}[(E_m)_j(E_m)_k] = \frac{1}{NT} [\Psi_{m,\sigma^2}]_{j,k}, $$

where $\Psi_{m,\sigma^2} \in \mathbb{R}^{m \times m}$ is defined in (4.1). Thus

$$ \mathbb{E}[M_1] = 2\mathbb{E}[E_m^\top \Psi_m^{-1} E_m] = 2\mathbb{E}[\mathrm{Tr}(\Psi_m^{-1} E_m E_m^\top)] \le \frac{2}{NT} \mathrm{Tr}(\Psi_m^{-1} \Psi_{m,\sigma^2}). $$

Collecting the bounds, we get

$$ \mathbb{E}[T_1] \le \inf_{h \in S_m} \|t - b_A\|_{\bar{f}_T}^2 + \frac{2}{NT} \mathrm{Tr}(\Psi_m^{-1} \Psi_{m,\sigma^2}). $$

### Remainder terms $T_2, T_3$. We use the following auxiliary bounds (proved in Section 7.3 and 7.3).

**Lemma 7.1.** Under the assumptions of Proposition 4.1, there exists $c > 0$ such that $\mathbb{P}(\Lambda_m^c) \le c(NT)^{-7}$ and $\mathbb{P}(\Omega_m^c) \le c(NT)^{-7}$.

**Lemma 7.2.** Under the assumptions of Proposition 4.1, there exists $c > 0$ such that $\mathbb{E}[(E_m^\top E_m)^2] \le 2c \frac{mL^2(m)}{(NT)^2} \int_{\mathbb{R}} \sigma_A^4(y) \bar{f}_T(y) dy$.

By the Cauchy-Schwarz inequality, Lemmas 7.1–7.2, and the stability condition (3.4),

$$ \begin{aligned} \mathbb{E}[T_2] & \le 2\mathbb{E}[\|\hat{b}_m - b_{A,m}\|_N^2 + \|b_{A,m} - b_A\|_N^2] \\ & \le 2\mathbb{E}[\|\hat{b}_m - b_{A,m}\|_N^2 + \|b_A\|_N^2] \\ & \le 2\mathbb{E}\left[(E_m^\top \hat{\Psi}_m^{-1} E_m + \|b_A\|_N^2) \mathbf{1}_{\Lambda_m \cap \Omega_m^c}\right] \\ & \le 2\mathbb{E}\left[(\|\hat{\Psi}_m^{-1}\|_{\mathrm{op}} E_m^\top E_m + \|b_A\|_N^2) \mathbf{1}_{\Lambda_m \cap \Omega_m^c}\right] \\ & \le \frac{c_T NT}{L(m) \log(NT)} \left(\mathbb{E}[(E_m^\top E_m)^2]\right)^{1/2} \left[\mathbb{P}(\Omega_m^c)\right]^{1/2} + 2\left(\mathbb{E}\|b_A\|_N^4\right)^{1/2} \left[\mathbb{P}(\Lambda_m^c)\right]^{1/2} \le \frac{C}{(NT)^3}, \end{aligned} \quad (7.13) $$

and analogously $\mathbb{E}[T_3] \le C(NT)^{-3}$. Combining the three pieces gives the stated bound in Proposition 4.1. $\square$

### Proof of Lemma 7.1.

Introduce the independent, centered, self-adjoint matrices

$$ K_{m,i} := \Psi_m^{-1/2} \tilde{K}_{m,i} \Psi_m^{-1/2}, \quad \tilde{K}_{m,i} := \frac{1}{NT} \int_0^T \delta_s^i \varphi^{(m)}(Y_s^i) \varphi^{(m)}(Y_s^i)^\top ds, $$
where $\varphi^{(m)}(y) = (\varphi_0(y), \dots, \varphi_{m-1}(y))^{\top}$. Then $Y := \sum_{i=1}^N K_{m,i} = \Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2}$ and $\mathbb{E}[K_{m,i}] = N^{-1} \mathrm{Id}_m$. Using $\|\varphi^{(m)}(y)\|_{2,m}^2 \le L(m)$,

$$ \|K_{m,i}\|_{\mathrm{op}} \le \|\Psi_m^{-1}\|_{\mathrm{op}} \frac{1}{NT} \int_0^T L(m) \, ds = \frac{\|\Psi_m^{-1}\|_{\mathrm{op}} L(m)}{N} =: L. $$

By (3.4), $L(m)(\|\Psi_m^{-1}\|_{\mathrm{op}} \vee 1) \le \frac{c_T T}{2 \log(NT)}$, hence $L \le \frac{c_T T}{2 \log(NT)}$. Set $c_{\varepsilon} := \frac{1}{2} - \frac{3}{2} \log(3/2)$. Matrix Chernoff (Lemma B.1) with $\varepsilon = \frac{1}{2}$ yields

$$ \mathbb{P}(\lambda_{\max}(Y) \ge \frac{3}{2}) \vee \mathbb{P}(\lambda_{\min}(Y) \le \frac{1}{2}) \le m \exp\left(\frac{c_{\varepsilon}}{L}\right). $$

Because $c_T = \frac{3 \log(3/2) - 1}{8T} = -\frac{c_{\varepsilon}}{4T}$, the bound on $L$ implies $\frac{c_{\varepsilon}}{L} \le -8 \log(NT)$, so

$$ \mathbb{P}(\|Y - \mathrm{Id}_m\|_{\mathrm{op}} \ge \frac{1}{2}) \le 2m (NT)^{-8} \le 2 (NT)^{-7}. $$

This is exactly $\mathbb{P}(\Omega_m^c) \le c(NT)^{-7}$. The bound for $\mathbb{P}(\Lambda_m^c)$ follows from Lemma A.2. $\square$

*Proof of Lemma 7.2.*

For $j = 0, \dots, m-1$ set

$$ M_j(T) := \sum_{i=1}^{N} \int_{0}^{T} \delta_{u}^{i} \varphi_{j}(Y_{u}^{i}) \sigma(Y_{u}^{i}) dW_{u}^{i}, \quad (E_{m})_{j} = \frac{M_{j}(T)}{NT}. $$

Let $F(x) = (\sum_{j=0}^{m-1} x_j^2)^2$. Since $F(x) \le m \sum_j x_j^4$,

$$ \mathbb{E}[(E_m^\top E_m)^2] = \frac{1}{N^4 T^4} \mathbb{E}[F(M_0, \dots, M_{m-1})] \le \frac{m}{N^4 T^4} \sum_{j=0}^{m-1} \mathbb{E}[M_j^4(T)]. $$

By Burkholder-Davis-Gundy Inequality (order 4), $\mathbb{E}[M_j^4(T)] \le c \mathbb{E}[\langle M_j \rangle_T^2]$, where

$$ \langle M_j \rangle_T = \sum_{i=1}^{N} \int_{0}^{T} \delta_{u}^{i} \varphi_{j}^{2}(Y_{u}^{i}) \sigma^{2}(Y_{u}^{i}) du. $$

Cauchy-Schwarz in time and in the sum over $i$ gives

$$ \mathbb{E}[\langle M_j \rangle_T^2] \le NT \sum_{i=1}^N \mathbb{E} \left[ \int_0^T \delta_u^i \varphi_j^4(Y_u^i) \sigma^4(Y_u^i) du \right]. $$

Since $\varphi_j^4(y) \le L(m) \varphi_j^2(y)$ and $\sum_{j=0}^{m-1} \varphi_j^2(y) \le L(m)$,

$$ \sum_{j=0}^{m-1} \mathbb{E}[\langle M_j \rangle_T^2] \le (NT)^2 L^2(m) \int_{\mathbb{R}} \sigma_A^4(y) \bar{f}_T(y) dy, $$

and the claim follows. $\square$

### 7.4. Proof of Theorem 4.3.

Define

$$ \nu_N(h) := \frac{1}{NT} \sum_{i=1}^{N} \int_{0}^{T} \delta_{s}^{i} t(Y_{s}^{i}) \sigma(X_{s}^{i}) dW_{s}^{i}, \qquad (7.14) $$
and

$$ \Omega_N := \bigcap_{m \in \mathcal{M}_N^+} \Omega_m, \quad \mathcal{M}_N^+ := \left\{ 1 \le m \le NT : c_{\varphi}^2 m (\|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le \frac{3c_T}{2} \frac{NT}{\log(NT)} \right\}. $$

By Lemma 7.1 and a union bound,

$$ \mathbb{P}(\Omega_N^c) \le c_1 (NT)^{-6}. \quad (7.15) $$

On $\Omega_N$, using Lemma A.1, we have the Löowner orders $\frac{2}{3}\Psi_m^{-1} \le \hat{\Psi}_m^{-1} \le 2\Psi_m^{-1}$ for all $m \in \mathcal{M}_N^+$, so by construction

$$ \mathcal{M}_N \subseteq \hat{\mathcal{M}}_N \subseteq \mathcal{M}_N^+. \quad (7.16) $$

**Risk decomposition.** Let $\hat{m}$ be defined by (4.7). Decompose

$$ \hat{b}_{\hat{m}} - b_A = (\hat{b}_{\hat{m}} - b_A)\mathbf{1}_{\Omega_N} + (\hat{b}_{\hat{m}} - b_A)\mathbf{1}_{\Omega_N^c} =: (\text{I}) + (\text{II}). $$

**Term (II).** One has $\hat{m} \le NT$, since $\hat{m} \in \hat{\mathcal{M}}_N$. As in (7.13), for every $m \ge 1$, $\|\hat{b}_m - b_A\|_N^2 \lesssim E_m^\top \hat{\Psi}_m^{-1} E_m + \|b_A\|_N^2$, hence, by the monotonicity of $m \mapsto E_m^\top E_m$ (it adds non-negative summands),

$$ \|\hat{b}_{\hat{m}} - b_A\|_N^2 \mathbf{1}_{\Omega_N^c} \lesssim \left( \sup_{1 \le m \le NT} \|\hat{\Psi}_m^{-1}\|_{\text{op}} \right) (E_{NT}^\top E_{NT}) \mathbf{1}_{\Omega_N^c} + \|b_A\|_N^2 \mathbf{1}_{\Omega_N^c}. $$

By the definition of $\hat{\mathcal{M}}_N$,

$$ \sup_{1 \le m \le NT} (\|\hat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le \frac{c_T}{c_{\varphi}^2} \frac{NT}{\log(NT)}. $$

Therefore, using Cauchy-Schwarz and then Lemma 7.2 with $m = NT$,

$$ \begin{aligned} \mathbb{E}[\|\hat{b}_{\hat{m}} - b_A\|_N^2 \mathbf{1}_{\Omega_N^c}] &\lesssim \frac{NT}{\log(NT)} (\mathbb{E}[(E_{NT}^\top E_{NT})^2])^{1/2} [\mathbb{P}(\Omega_N^c)]^{1/2} + (\mathbb{E}\|b_A\|_N^4)^{1/2} [\mathbb{P}(\Omega_N^c)]^{1/2} \\ &\lesssim \frac{NT}{\log(NT)} \left( \frac{NT L^2(NT)}{(NT)^2} \int_{\mathbb{R}} \sigma_A^4(y) \bar{f}_T(y) dy \right)^{1/2} [\mathbb{P}(\Omega_N^c)]^{1/2} + \frac{1}{(NT)^3}, \end{aligned} $$

where we also used (7.15) for the last term. Since $L(m) \le c_{\varphi}^2 m$, the first term is bounded by a constant multiple of $(NT)^{-1}$. Consequently,

$$ \mathbb{E}[\|\hat{b}_{\hat{m}} - b_A\|_N^2 \mathbf{1}_{\Omega_N^c}] \le \frac{C}{NT}. \quad (7.17) $$

**Term (I).** By the definition of $\hat{m}$ (4.7), for every $m \in \hat{\mathcal{M}}_N$,

$$ -\|\hat{b}_{\hat{m}}\|_N^2 + \text{pen}(\hat{m}) \le -\|\hat{b}_m\|_N^2 + \text{pen}(m). $$

Since we have the inclusion (7.16), the previous relation holds for every $m \in \mathcal{M}_N$. Equivalently, by (4.6),

$$ \gamma_N(\hat{b}_{\hat{m}}) + \text{pen}(\hat{m}) \le \gamma_N(\hat{b}_m) + \text{pen}(m). \quad (7.18) $$

Fix $m^* \in \mathcal{M}_N$ and $h^* \in S_{m^*}$ by 7.18, and the definition of $\hat{b}_{m^*}$ (3.1).

$$ \gamma_N(\hat{b}_{\hat{m}}) + \text{pen}(\hat{m}) \le \gamma_N(\hat{b}_{m^*}) + \text{pen}(m^*) \le \gamma_N(h^*) + \text{pen}(m^*) \quad (7.19) $$

From (7.19) and $\gamma_N(\hat{b}_{\hat{m}}) - \gamma_N(h^*) = \|\hat{b}_{\hat{m}} - b_A\|_N^2 - \|h^* - b_A\|_N^2 + 2\nu_N(\hat{b}_{\hat{m}} - h^*)$, we deduce

$$ \|\hat{b}_{\hat{m}} - b_A\|_N^2 \le \|h^* - b_A\|_N^2 + 2\nu_N(\hat{b}_{\hat{m}} - h^*) + \text{pen}(m^*) - \text{pen}(\hat{m}). \quad (7.20) $$
Using $2ab \le \frac{1}{8}a^2 + 8b^2$ with $a = \|\hat{b}_m - b_A\|_{\bar{f}_T}$ and $b = \sup_{t \in B_{\hat{m},m^*}^{\bar{f}_T}(0,1)} \nu_N(h)$ and the triangle inequality,

$$ \begin{aligned} \|\hat{b}_m - b_A\|_N^2 &\le \|h^* - b_A\|_N^2 + 8 \sup_{t \in B_{\hat{m},m^*}^{\bar{f}_T}(0,1)} \nu_N^2(h) + \frac{1}{8}\|\hat{b}_m - h^*\|_{\bar{f}_T}^2 + \text{pen}(m^*) - \text{pen}(\hat{m}) \\ &\le \|h^* - b_A\|_N^2 + 8 \sup_{t \in B_{\hat{m},m^*}^{\bar{f}_T}(0,1)} \nu_N^2(h) + \frac{1}{4}\|\hat{b}_m - b_A\|_N^2 + \frac{1}{4}\|h^* - b_A\|_N^2 + \text{pen}(m^*) - \text{pen}(\hat{m}), \end{aligned} $$

since $\|u\|_{\bar{f}_T}^2 \le 2\|u\|_N^2$ on $\Omega_N$. Rearranging,

$$ \begin{aligned} \|\hat{b}_m - b_A\|_N^2 &\le \frac{5}{3}\|h^* - b_A\|_N^2 + \frac{32}{3} \sup_{t \in B_{\hat{m},m^*}^{\bar{f}_T}(0,1)} \nu_N^2(h) + \frac{4}{3}\text{pen}(m^*) - \frac{4}{3}\text{pen}(\hat{m}) \\ &\le \frac{5}{3}\|h^* - b_A\|_N^2 + \frac{32}{3} \left( \sup_{t \in B_{\hat{m},m^*}^{\bar{f}_T}(0,1)} \nu_N^2(h) - p(\hat{m},m^*) \right) + \frac{32}{3}p(\hat{m},m^*) \quad (7.21) \\ &\quad + \frac{4}{3}\text{pen}(m^*) - \frac{4}{3}\text{pen}(\hat{m}). \quad (7.22) \end{aligned} $$

Set, for $m, m' \le NT$,

$$ p(m, m') := \tau \frac{c_1^2(m + m')}{NT}, \quad \tau > 0. $$

The following lemma is proved in Appendix C.

**Lemma 7.3.** There exist numerical constants $\tau = 38$ and $c > 0$ such that,

$$ \mathbb{E}\left[\left(\sup_{t \in B_{\hat{m},m^*}^{\bar{f}_T}(0,1)} \nu_N^2(h) - p(m,m')\right) + \mathbf{1}_{\Omega_N}\right] \le c c_1^2 \frac{1}{NT}. $$

Choose $\kappa \ge 8\tau =: \kappa_0$ in $\text{pen}(k) = \kappa c_1^2/(NT)$. Taking expectations in (7.21), applying Lemma 7.3, minimizing over $m^* \in \mathcal{M}_N$ and $h^* \in S_{m^*}$, and adding (7.17), we obtain

$$ \mathbb{E}[\|\hat{b}_m - b_A\|_N^2] \le \frac{8}{3} \inf_{m \in \mathcal{M}_N} \left\{ \inf_{h \in S_m} \|t - b_A\|_{\bar{f}_T}^2 + \text{pen}(m) \right\} + \frac{C}{NT}, $$

which proves Theorem 4.3. $\square$

## Appendix A. Algebraic facts about $\Omega_m$ and $\Lambda_m$

$$ \text{Fix } m \in \mathbb{N} \text{ and let } \Psi_m = (\langle \varphi_j, \varphi_k \rangle_{\bar{f}_T})_{0 \le j, k < m}. $$

**Lemma A.1.** Assume $\Psi_m$ is invertible and define $\Omega_m$ by (7.12). Then

$$ \Omega_m = \left\{ \| \Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} - \text{Id}_m \|_{\text{op}} \le \frac{1}{2} \right\}. $$

Moreover, on $\Omega_m$ all eigenvalues of

$$ R_m := \Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} $$

lie in the interval $[\frac{1}{2}, \frac{3}{2}]$.

*Proof:* For $t = \sum_{j=0}^{m-1} \beta_j \varphi_j$ with $\beta \in \mathbb{R}^m$, $\|t\|_N^2 = \beta^\top \hat{\Psi}_m \beta$ and $\|h\|_{\bar{f}_T}^2 = \beta^\top \Psi_m \beta$. Setting $u := \Psi_m^{1/2} \beta$ gives

$$ \left| \frac{\|t\|_N^2}{\|h\|_{\bar{f}_T}^2} - 1 \right| = \frac{|u^\top (\Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} - \text{Id}_m) u|}{\|u\|_2^2} = \|R_m - \text{Id}_m\|_{\text{op}}. $$
This proves the stated equivalence.

For the eigenvalue assertion, note that if $\|R_m - \mathrm{Id}_m\|_{\mathrm{op}} \le 1/2$, then every eigenvalue $\lambda$ of $R_m$ satisfies $|\lambda - 1| \le 1/2$. □

**Lemma A.2.** Let

$$ \Lambda_m := \left\{ L(m) \left( \| \hat{\Psi}_m^{-1} \|_{\mathrm{op}} \vee 1 \right) \le c_T \frac{NT}{\log(NT)} \right\}, \quad L(m) = \sup_{x \in A} \sum_{j=0}^{m-1} \varphi_j^2(x). $$

Under (3.4), $\Omega_m \subset \Lambda_m$.

*Proof:* On $\Omega_m$, the eigenvalues of $R_m := \Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} - \mathrm{Id}_m$ lie in $[-\frac{1}{2}, \frac{1}{2}]$; hence $\frac{2}{3} \mathrm{Id}_m \le (\mathrm{Id}_m + R_m)^{-1} \le 2 \mathrm{Id}_m$. Multiplying by $\Psi_m^{-1/2}$ yields $\frac{2}{3} \Psi_m^{-1} \le \hat{\Psi}_m^{-1} \le 2 \Psi_m^{-1}$, and the claim follows from (3.4). □

## Appendix B. Auxiliary probabilistic inequalities

**Lemma B.1 (Matrix Chernoff Tropp et al. (2015)).** Let $X_1, \dots, X_N$ be independent, self-adjoint $d \times d$ random matrices with $0 \preceq X_i \preceq L$ Id. Set $Y := \sum_{i=1}^N X_i$ and $\mu_{\min} := \lambda_{\min}(EY)$, $\mu_{\max} := \lambda_{\max}(EY)$. For every $\varepsilon \in (0, 1]$,

$$ \begin{aligned} \mathbb{P}(\lambda_{\min}(Y) \le (1 - \varepsilon)\mu_{\min}) &\le d \left[ \frac{e^{-\varepsilon}}{(1-\varepsilon)^{1-\varepsilon}} \right]^{\mu_{\min}/L}, \\ \mathbb{P}(\lambda_{\max}(Y) \ge (1 + \varepsilon)\mu_{\max}) &\le d \left[ \frac{e^{\varepsilon}}{(1+\varepsilon)^{1+\varepsilon}} \right]^{\mu_{\max}/L}. \end{aligned} $$

**Lemma B.2 (Bernstein for continuous martingales).** Let $(M_t)_{t \ge 0}$ be a continuous local martingale with quadratic variation $\langle M \rangle_t$. For any $a, b > 0$,

$$ \mathbb{P}\left(\sup_{t \ge 0} |M_t| \ge a, \langle M \rangle_{\infty} \le b\right) \le 2 \exp(-a^2/(2b)). $$

**Lemma B.3 (Finite $\delta$-nets in $D$-dimensional spaces).** Let $S \subset L^2(d\mu)$ be a linear subspace of dimension $D$. For every $0 < \delta < 1$ there exists a finite set $T_\delta \subset S$ such that

$$ |T_{\delta}| \le \left(\frac{3}{\delta}\right)^{D}, \quad \forall t \in S, \quad \|t\| \le 1 \ \exists s \in T_{\delta}: \ \|t-s\| \le \delta. $$

## Appendix C. Proof of Lemma 7.3

We give a detailed chaining argument for the quadratic empirical process.

$$ \nu_N(t) = \frac{1}{NT} \sum_{i=1}^{N} \int_{0}^{T} \delta_s^i t(Y_s^i) \sigma(X_s^i) dW_s^i, \quad t \in S_m + S_{m'}. $$

Throughout, set $S := S_m + S_{m'}$ and $D := \dim(S) \le m + m'$. Let $\mathbb{B} := \{t \in S : \|t\|_{\bar{f}_T} \le 1\}$. On $\Omega_N$ we have the norm equivalence

$$ \forall t \in S, \quad \frac{1}{2} \|h\|_{\bar{f}_T}^2 \le \|t\|_N^2 \le \frac{3}{2} \|h\|_{\bar{f}_T}^2. \qquad (C.1) $$

For fixed $t \in S$, $(M_t)_{t \ge 0} := \sum_{i=1}^N \int_0^u \delta_s^i t(Y_s^i) \sigma(X_s^i) dW_s^i$ is a continuous martingale with

$$ \langle M_t \rangle_T = \sum_{i=1}^{N} \int_{0}^{T} \delta_{s}^{i} t^{2}(Y_{s}^{i}) \sigma^{2}(X_{s}^{i}) ds \leq N T c_{1}^{2} \|h\|_{\bar{f}_{T}}^{2}. $$
Applying the Bernstein inequality for continuous martingales (Lemma B.2) and using (C.1), we obtain, for all $x > 0$,

$$ \mathbb{P}(\nu_N(t) \ge \sqrt{3} c_1 \|t\|_{\bar{f}_T} \sqrt{x}, \Omega_N) \le e^{-NTx}. \quad (C.2) $$

For $\delta \in (0, 1)$, let $T_\delta \subset \mathbb{B}$ be a $\delta$-net of $\mathbb{B}$ in $\|\cdot\|_{\bar{f}_T}$ with cardinality $|T_\delta| \le (3/\delta)^D$ (Lemma B.3). Fix a base mesh $\delta_0 \in (0, 1)$ and define $\delta_k := \delta_0 2^{-k}$ and $H_k := \ln|T_{\delta_k}| \le D \ln(3/\delta_k)$. By a standard telescoping approximation (see, e.g., Lorentz et al. (1996, Chap. 15)), every $t \in \mathbb{B}$ admits a representation

$$ t = t_0 + \sum_{k \ge 1} (t_k - t_{k-1}), \quad t_k \in T_{\delta_k}, $$

with

$$ \|t_k - t_{k-1}\|_{\bar{f}_T} \le \|t - t_{k-1}\|_{\bar{f}_T} + \|t - t_k\|_{\bar{f}_T} \le \delta_{k-1} + \delta_k \le \frac{3}{2} \delta_{k-1}. \quad (C.3) $$

In particular,

$$ \|t_k - t_{k-1}\|_{\bar{f}_T}^2 \le \frac{9}{4} \delta_{k-1}^2 \le \frac{5}{2} \delta_{k-1}^2. \quad (C.4) $$

Let $x_0 > 0$ and $x_k > 0$ be parameters to be chosen. By a union bound over $t_0 \in T_{\delta_0}$ and pairs $(t_{k-1}, t_k) \in T_{\delta_{k-1}} \times T_{\delta_k}$, and using (C.2) with $t$ replaced by $t_0$ and $t_k - t_{k-1}$, we obtain

$$ \mathbb{P}(\sup_{t \in \mathbb{B}} \nu_N(h) > \Delta, \Omega_N) \le e^{H_0 - NTx_0} + \sum_{k \ge 1} e^{H_{k-1} + H_k - NTx_k}, \quad (C.5) $$

where

$$ \Delta := \sqrt{3} c_1 \left[ \sqrt{x_0} + \sum_{k \ge 1} \|t_k - t_{k-1}\|_{\bar{f}_T} \sqrt{x_k} \right]. \quad (C.6) $$

Using (C.4), this will be further upper bounded by $\Delta \le \sqrt{3} c_1 [\sqrt{x_0} + \sum_{k \ge 1} \delta_{k-1} \sqrt{\frac{5}{2} x_k}]$.

Choose

$$ NT x_0 := H_0 + m' + u, \quad NT x_k := H_{k-1} + H_k + k + m' + u \quad (k \ge 1), \quad (C.7) $$

with $u > 0$. Plugging (C.7) into (C.5) yields

$$ \mathbb{P}(\sup_{t \in \mathbb{B}} \nu_N(h) > \Delta, \Omega_N) \le e^{-(m'+u)} + \sum_{k \ge 1} e^{-(k+m'+u)} \le 1.6 e^{-(m'+u)}. \quad (C.8) $$

Since $H_k \le D \ln(3/\delta_k) = D(\ln(3/\delta_0) + k \ln 2)$, we have

$$ x_0 = \frac{H_0 + m' + u}{NT} \le \frac{D \ln(3/\delta_0) + m' + u}{NT}, $$

and, for $k \ge 1$,

$$ x_k = \frac{H_{k-1} + H_k + k + m' + u}{NT} \le \frac{2D \ln(3/\delta_0) + 2Dk \ln 2 + k + m' + u}{NT}. $$

Using $\delta_{k-1} = \delta_0 2^{-(k-1)}$ and the elementary bound $\sum_{k \ge 1} 2^{-(k-1)} \sqrt{a+bk} \le C_* \sqrt{a+b}$ valid for all $a, b \ge 0$ (for some universal $C_* > 0$), we infer that, for a numerical constant $c_0 > 0$,

$$ \Delta^2 \le \tau c_1^2 \frac{D + m' + u}{NT}, \quad \text{with } \tau = \tau(\delta_0) := 3 \left(1 + \frac{5}{2} c_0^2 \delta_0^2\right). \quad (C.9) $$

A direct (but tedious) check shows that with $\delta_0 = 0.0138$ one can take $\tau = 38$. (Any explicit choice with $\tau$ finite suffices for the theorem; we retain $\tau = 38$ to match the statement.)

Let

$$ p(m, m') := \tau c_1^2 \frac{m + m'}{NT}. $$
Combining (C.8)–(C.9) and using that $D \le m + m'$, we get, for every $u > 0$,

$$ \mathbb{P}\left(\sup_{t \in \mathbb{B}} \nu_N^2(h) > p(m, m') + \tau c_1^2 \frac{u}{NT}, \Omega_N\right) \le 1.6 e^{-(m'+u)}. $$

Integrating this tail bound over $u \in (0, \infty)$ (i.e., using $\mathbb{E}[(X-a)_+] = \int_0^\infty \mathbb{P}(X>a+v) dv$) yields

$$ \mathbb{E}\left[(\sup_{t \in \mathbb{B}} \nu_N^2(h) - p(m, m'))_+ \mathbf{1}_{\Omega_N}\right] \le c c_1^2 \frac{e^{-m'}}{NT}, $$

for a numerical constant $c > 0$. This is exactly the claim of Lemma 7.3 (after renaming constants). □

<figure><img src="image_13.png" /></figure>

## Appendix D. Simulation Tables

<table><caption>TABLE D.2. Censoring rates (mean ± sd) across models, censoring modes, and parameters. Only the relevant tuning parameter is shown ($c_0$ for constant, $\tau$ for OU-censor and driftedBM). Settings: $K = 100$, $N = 50$, $T = 5$, $\Delta t = 0.01$, $x_0 = 0$.</caption><thead><tr><th>Model</th><th>Censoring mode</th><th>Parameter</th><th>1 - δ̂<sub>T</sub> (mean ± sd)</th></tr></thead><tbody><tr><td rowspan="7">OU</td><td>constant</td><td>c<sub>0</sub> = 0.5</td><td>0.775 ± 0.024</td></tr><tr><td>constant</td><td>c<sub>0</sub> = 1.0</td><td>0.932 ± 0.012</td></tr><tr><td>constant</td><td>c<sub>0</sub> = 1.5</td><td>0.986 ± 0.006</td></tr><tr><td>OU-censor</td><td>τ = 0.3</td><td>0.924 ± 0.016</td></tr><tr><td>OU-censor</td><td>τ = 0.5</td><td>0.910 ± 0.017</td></tr><tr><td>OU-censor</td><td>τ = 0.7</td><td>0.887 ± 0.020</td></tr><tr><td>driftedBM</td><td>τ = 0.3</td><td>0.893 ± 0.022</td></tr><tr><td rowspan="7">Double-well</td><td>constant</td><td>c<sub>0</sub> = 0.5</td><td>0.663 ± 0.052</td></tr><tr><td>constant</td><td>c<sub>0</sub> = 1.0</td><td>0.856 ± 0.024</td></tr><tr><td>constant</td><td>c<sub>0</sub> = 1.5</td><td>0.992 ± 0.003</td></tr><tr><td>OU-censor</td><td>τ = 0.3</td><td>0.852 ± 0.027</td></tr><tr><td>OU-censor</td><td>τ = 0.5</td><td>0.843 ± 0.026</td></tr><tr><td>OU-censor</td><td>τ = 0.7</td><td>0.829 ± 0.027</td></tr><tr><td>driftedBM</td><td>τ = 0.3</td><td>0.841 ± 0.028</td></tr><tr><td rowspan="7">CIR</td><td>constant</td><td>c<sub>0</sub> = 0.5</td><td>0.649 ± 0.022</td></tr><tr><td>constant</td><td>c<sub>0</sub> = 1.0</td><td>0.928 ± 0.011</td></tr><tr><td>constant</td><td>c<sub>0</sub> = 1.5</td><td>0.988 ± 0.004</td></tr><tr><td>OU-censor</td><td>τ = 0.3</td><td>0.910 ± 0.015</td></tr><tr><td>OU-censor</td><td>τ = 0.5</td><td>0.875 ± 0.019</td></tr><tr><td>OU-censor</td><td>τ = 0.7</td><td>0.834 ± 0.021</td></tr><tr><td>driftedBM</td><td>τ = 0.3</td><td>0.839 ± 0.029</td></tr><tr><td rowspan="7">CIR</td><td>constant</td><td>c<sub>0</sub> = 0.5</td><td>0.649 ± 0.022</td></tr><tr><td>constant</td><td>c<sub>0</sub> = 1.0</td><td>0.928 ± 0.011</td></tr><tr><td>constant</td><td>c<sub>0</sub> = 1.5</td><td>0.988 ± 0.004</td></tr><tr><td>OU-censor</td><td>τ = 0.3</td><td>0.910 ± 0.015</td></tr><tr><td>OU-censor</td><td>τ = 0.5</td><td>0.875 ± 0.019</td></tr><tr><td>OU-censor</td><td>τ = 0.7</td><td>0.834 ± 0.021</td></tr><tr><td>driftedBM</td><td>τ = 0.3</td><td>0.839 ± 0.029</td></tr></tbody></table>

## References

* Abramowitz, M. and Stegun, I. A. *Handbook of mathematical functions with formulas, graphs, and mathematical tables*, volume 55. US Government printing office (1968).

* Aubin, J.-P. and Da Prato, G. Stochastic viability and invariance. *Annali della Scuola Normale Superiore di Pisa-Classe di Scienze*, **17** (4), 595–613 (1990).

* Beran, R. Nonparametric regression with randomly censored survival data (1981).
<table><caption>TABLE D.3. OU model, trigonometric basis (compact A).</caption><thead><tr><th>NT</th><th>Uncensored</th><th>Constant</th><th>OU-censor</th><th>DriftedBM</th></tr></thead><tbody><tr><td>10</td><td>2.983×10<sup>-1</sup> ± 2.3×10<sup>-1</sup></td><td>2.710×10<sup>-1</sup> ± 3.0×10<sup>-1</sup></td><td>3.405×10<sup>-1</sup> ± 2.8×10<sup>-1</sup></td><td>3.023×10<sup>-1</sup> ± 2.3×10<sup>-1</sup></td></tr><tr><td>100</td><td>8.543×10<sup>-2</sup> ± 4.2×10<sup>-2</sup></td><td>1.374×10<sup>-1</sup> ± 5.8×10<sup>-2</sup></td><td>1.199×10<sup>-1</sup> ± 5.1×10<sup>-2</sup></td><td>1.053×10<sup>-1</sup> ± 5.7×10<sup>-2</sup></td></tr><tr><td>500</td><td>2.682×10<sup>-2</sup> ± 1.2×10<sup>-2</sup></td><td>1.151×10<sup>-1</sup> ± 2.5×10<sup>-2</sup></td><td>5.937×10<sup>-2</sup> ± 1.6×10<sup>-2</sup></td><td>4.304×10<sup>-2</sup> ± 2.0×10<sup>-2</sup></td></tr><tr><td>2000</td><td>5.886×10<sup>-3</sup> ± 2.7×10<sup>-3</sup></td><td>1.029×10<sup>-1</sup> ± 9.7×10<sup>-3</sup></td><td>3.945×10<sup>-2</sup> ± 7.1×10<sup>-3</sup></td><td>1.598×10<sup>-2</sup> ± 6.8×10<sup>-3</sup></td></tr><tr><td>5000</td><td>2.580×10<sup>-3</sup> ± 1.1×10<sup>-3</sup></td><td>9.784×10<sup>-2</sup> ± 7.0×10<sup>-3</sup></td><td>3.445×10<sup>-2</sup> ± 4.5×10<sup>-3</sup></td><td>8.417×10<sup>-3</sup> ± 3.5×10<sup>-3</sup></td></tr><tr><td>10000</td><td>1.318×10<sup>-3</sup> ± 5.4×10<sup>-4</sup></td><td>9.817×10<sup>-2</sup> ± 4.2×10<sup>-3</sup></td><td>3.331×10<sup>-2</sup> ± 3.3×10<sup>-3</sup></td><td>5.743×10<sup>-3</sup> ± 2.0×10<sup>-3</sup></td></tr></tbody></table>

<table><caption>TABLE D.4. Double-well model, Hermite basis (non-compact).</caption><thead><tr><th>NT</th><th>Uncensored</th><th>Constant</th><th>OU-censor</th><th>DriftedBM</th></tr></thead><tbody><tr><td>10</td><td>1.151×10<sup>-1</sup> ± 9.7×10<sup>-2</sup></td><td>9.989×10<sup>-2</sup> ± 9.8×10<sup>-2</sup></td><td>1.275×10<sup>-1</sup> ± 1.2×10<sup>-1</sup></td><td>1.224×10<sup>-1</sup> ± 1.0×10<sup>-1</sup></td></tr><tr><td>100</td><td>6.652×10<sup>-2</sup> ± 2.0×10<sup>-2</sup></td><td>7.982×10<sup>-2</sup> ± 3.1×10<sup>-2</sup></td><td>7.763×10<sup>-2</sup> ± 2.7×10<sup>-2</sup></td><td>6.619×10<sup>-2</sup> ± 2.1×10<sup>-2</sup></td></tr><tr><td>500</td><td>9.347×10<sup>-3</sup> ± 4.2×10<sup>-3</sup></td><td>6.841×10<sup>-2</sup> ± 1.4×10<sup>-2</sup></td><td>3.970×10<sup>-2</sup> ± 1.1×10<sup>-2</sup></td><td>1.610×10<sup>-2</sup> ± 7.3×10<sup>-3</sup></td></tr><tr><td>2000</td><td>1.879×10<sup>-3</sup> ± 9.2×10<sup>-4</sup></td><td>6.463×10<sup>-2</sup> ± 8.0×10<sup>-3</sup></td><td>3.267×10<sup>-2</sup> ± 5.0×10<sup>-3</sup></td><td>5.755×10<sup>-3</sup> ± 2.4×10<sup>-3</sup></td></tr><tr><td>5000</td><td>9.714×10<sup>-4</sup> ± 3.8×10<sup>-4</sup></td><td>6.400×10<sup>-2</sup> ± 4.7×10<sup>-3</sup></td><td>3.183×10<sup>-2</sup> ± 3.7×10<sup>-3</sup></td><td>3.412×10<sup>-3</sup> ± 1.5×10<sup>-3</sup></td></tr><tr><td>10000</td><td>5.091×10<sup>-4</sup> ± 2.2×10<sup>-4</sup></td><td>6.381×10<sup>-2</sup> ± 3.0×10<sup>-3</sup></td><td>3.161×10<sup>-2</sup> ± 2.3×10<sup>-3</sup></td><td>2.368×10<sup>-3</sup> ± 8.5×10<sup>-4</sup></td></tr></tbody></table>

<table><caption>TABLE D.5. CIR model, Laguerre basis ($\mathbb{R}_+$).</caption><thead><tr><th>NT</th><th>Uncensored</th><th>Constant</th><th>OU-censor</th><th>DriftedBM</th></tr></thead><tbody><tr><td>10</td><td>7.960×10<sup>-2</sup> ± 3.1×10<sup>-2</sup></td><td>6.602×10<sup>-2</sup> ± 3.2×10<sup>-2</sup></td><td>8.989×10<sup>-2</sup> ± 5.6×10<sup>-2</sup></td><td>7.550×10<sup>-2</sup> ± 4.3×10<sup>-2</sup></td></tr><tr><td>100</td><td>3.211×10<sup>-2</sup> ± 2.4×10<sup>-2</sup></td><td>6.146×10<sup>-2</sup> ± 3.4×10<sup>-2</sup></td><td>5.903×10<sup>-2</sup> ± 3.2×10<sup>-2</sup></td><td>4.138×10<sup>-2</sup> ± 3.1×10<sup>-2</sup></td></tr><tr><td>500</td><td>1.205×10<sup>-2</sup> ± 9.7×10<sup>-3</sup></td><td>6.326×10<sup>-2</sup> ± 1.3×10<sup>-2</sup></td><td>4.584×10<sup>-2</sup> ± 1.5×10<sup>-2</sup></td><td>1.686×10<sup>-2</sup> ± 7.4×10<sup>-3</sup></td></tr><tr><td>2000</td><td>3.008×10<sup>-3</sup> ± 1.7×10<sup>-3</sup></td><td>6.387×10<sup>-2</sup> ± 8.1×10<sup>-3</sup></td><td>4.399×10<sup>-2</sup> ± 7.3×10<sup>-3</sup></td><td>8.116×10<sup>-3</sup> ± 4.7×10<sup>-3</sup></td></tr><tr><td>5000</td><td>1.941×10<sup>-3</sup> ± 9.9×10<sup>-4</sup></td><td>7.558×10<sup>-2</sup> ± 5.1×10<sup>-3</sup></td><td>4.179×10<sup>-2</sup> ± 5.1×10<sup>-3</sup></td><td>4.397×10<sup>-3</sup> ± 2.0×10<sup>-3</sup></td></tr><tr><td>10000</td><td>1.119×10<sup>-3</sup> ± 6.3×10<sup>-4</sup></td><td>7.466×10<sup>-2</sup> ± 3.3×10<sup>-3</sup></td><td>4.139×10<sup>-2</sup> ± 3.3×10<sup>-3</sup></td><td>3.249×10<sup>-3</sup> ± 1.2×10<sup>-3</sup></td></tr></tbody></table>

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