Generating such grids numerically can be done choosing a starting point $x_0$ and adding points $x_j$ to the grid iteratively as follows: given $x_{j-1}$, let $x_j$ be chosen such that:

$$ \left| (s(y) - s(x_{j-1})) \int_{x_{j-1}}^{y} \frac{2}{s'(\zeta)\sigma^2(\zeta)} d\zeta \right| \le h^2/2. \quad (5.15) $$

Then the next element of the grid is defined as

$$ x_j = \begin{cases} x_j^{(0)} & \text{if } x_j^{(0)} - x_{j-1} \le h, \\ x_{j-1} + h & \text{if } x_j^{(0)} - x_{j-1} > h. \end{cases} \quad (5.16) $$

The problem (5.15) can be solved numerically using a fixed point algorithm. Examples of STMCA simulations using tuned grids computed solving (5.15)-(5.16) are given in Figures 6.2 and 6.3.

### 5.2.4 The case of diffusions with boundary conditions

When presenting the results and the structure of the scheme, we considered only processes where $\mathbb{I}$ is an open set, thus considering diffusion with unreachable boundaries. Our results also adapt to the situation where either $\ell$ and/or $r$ are reachable, and in this case some adjustments are needed, depending on the nature of finite boundaries and on the condition at regular boundaries. In order to keep the presentation simple, we assume that the process is on natural scale and that $\mathbb{I} = [0, +\infty)$ (the adaptation to $\mathbb{I} = (\ell, r]$ or $\mathbb{I} = [\ell, r]$ or $\mathbb{I} = [\ell, r)$ with $\ell \in \mathbb{R} \cup \{-\infty\}$ and $r \in \mathbb{R} \cup \{+\infty\}$ is straightforward).

It is well known (see for instance Section 5.11 of Itô's book [52]) that the finite boundary 0 can be of four types. Setting, for some fixed $c > 0$,

$$ \mathcal{I} = \int \int_{0<y<x<c} m(dx) dy, \quad \mathcal{II} = \int \int_{0<y<x<c} m(dy) dx, $$

then

* $0$ is an exit boundary if $\mathcal{I} < \infty$ and $\mathcal{II} = \infty$,

* $0$ is a regular boundary if $\mathcal{I} < \infty$ and $\mathcal{II} < \infty$,

* $0$ is a natural boundary if $\mathcal{I} = \infty$ and $\mathcal{II} = \infty$,

* $0$ is an entrance boundary if $\mathcal{I} = \infty$ and $\mathcal{II} < \infty$.

The entrance type been excluded for a finite boundary of a diffusion process on natural scale, and the natural type been considered in the settings of Theorem 5.2.1, this leaves us with two possible types for the boundary 0: exit or regular. If 0 is an exit boundary, then the diffusion process $X$ is absorbed at the boundary 0. If 0 is a regular boundary, then the diffusion process can either be absorbed or reflected at 0. In these cases, the convergence result of Theorem 5.2.1 can be extended by considering a grid $g$ on $\mathbb{I}$ containing 0 and by adapting the dynamics of $\tilde{X}^g$ as follows. The dynamic of $\tilde{X}^g$ is the same as in Algorithm 1, up to the time when it reaches 0, then:

* if 0 is an absorbing boundary (exit or regular), then the result can be immediately extended by stopping $\tilde{X}^g$ when it reaches 0;
<figure><img src="image_13.png" /><figcaption>Figure 5.3: The covering grid $\mathbf{g} = \mathbb{Z}^+$ of $[0, \infty)$ along with its first cells $C(\mathbf{g})$.</figcaption></figure>

• if 0 is a reflecting regular boundary, then the process $\tilde{X}^g$ jumps from 0 to $b := \min \mathbf{g} \setminus \{0\}$ with probability 1 and after a time $\int_{[0,b]} (b-\zeta) m(d\zeta)$. We emphasize that in this configuration, 0 may be a sticky boundary (i.e., with $m(0) \in (0, +\infty)$).

In both cases of reflection and absorption at 0, the boundary is attainable. Thus, 0 must be a point of any covering grid of $\mathbb{I} = [0, \infty)$. This leads to the following adaptation of the notion of grid cells. The cells $C(\mathbf{g})$ of such grid $\mathbf{g}$ are the open intervals for the induced topology of $\mathbb{R}$ on $\mathbb{I}$ with endpoint in $\mathbf{g}$. For example $\mathbf{g}_0 = \mathbb{Z}_+$ is a covering grid of $[0, \infty)$ and $C(\mathbf{g}_0) = \{[0, 1), (n-1, n+1)_{n \in \mathbb{N}}\}$.

The proof of the convergence in these situations is omitted here, since it is a straightforward adaptation of the proof of Theorem 5.2.1, using in particular the fact that, in the case of a reflecting boundary,

$$E_0[\tau_b] = \int_{[0,b)} (b - \zeta) m(d\zeta).$$

The case of killing boundaries, and in general of a process with non-zero killing measure, leads to additional non-trivial difficulties. Devising an algorithm and a similar result as Theorem 5.2.1 for such processes remains an active area of research.

### 5.2.5 Markovian embedding

The consecutive values of the process $(\tilde{X}_t)_{t \ge 0}$ defined through (5.2)-(5.4) form a Markov chain with, by construction, the same transition probabilities as $(X_t)_{t \ge 0}$ on $\mathbf{g}$. We define the embedding times of $(X_t)_{t \ge 0}$ in $\mathbf{g}$ as,

$$\left\{ \begin{array}{l} \tau_0^g = 0, \\ \tau_k^g = \inf \{ t > \tau_{k-1}^g : X_t \in \mathbf{g} \setminus \{ X_{\tau_{k-1}^g} \} \}, \quad \forall k \ge 1. \end{array} \right. \qquad (5.17)$$

As both $\tilde{X}_{Tg(n)}$ and $X_{\tau_n^g}$ are both Markov chains with the same transition probabilities with $\tilde{X}_0$ forced to be equal in law to $X_{\tau_1^g}$ (see Section 5.2.1), the following equality in law holds,

$$\mathrm{Law}(\tilde{X}_{Tg(n)}; n \ge 0) = \mathrm{Law}(X_{\tau_n^g}; n \ge 1).$$

We define $K^g(t)$ as the inverse of $T^g(n)$, i.e.,

$$K^g(t) = \inf \left\{ n \in \mathbb{N} : \sum_{k=1}^{n} \mathrm{E} \left[ \tau_k^g - \tau_{k-1}^g \mid X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] > t \right\}. \qquad (5.18)$$

Thus, we get the following Proposition.
**Proposition 5.2.7.** Let $(X_t)_{t \ge 0}$ be a diffusion process, $g$ a grid defined over its state space $\mathbb{I}$ and $\tilde{X}_t^g$ be the approximation process defined in (5.3) and (5.4). Then, if $(\tau_n^g)_{n \ge 0}$ are the embedding times of $(X_t)_{t \ge 0}$ in $g$, the following equality in law holds,

$$ \mathrm{Law}(\tilde{X}_t; t \ge 0) = \mathrm{Law}(X_{\tau_{Kg(t)}}; t \ge 0), $$

where $K^g(t)$ is the random index defined in (5.18).

## 5.3 Moment characterization of conditional exit times

The law of the approximation process defined in the previous section was shown to be determined by the transition probabilities $P_x(\tau_b < \tau_a)$ and conditional transition times $E_x(\tau_{ab}|\tau_b < \tau_a)$. In this section we show that quantities of the form $v_k(x) = E_x(\tau_{ab}^k 1_{\tau_b < \tau_a})$ yield an integral formulation with respect to the speed measure of the diffusion and involving the scale function (we do not assume that the diffusion is on natural scale in the present section). We also show that this results in them being solutions to Dirichlet problems where the differential operator is the infinitesimal generator $L$ of the diffusion. This allows us to simulate such processes via Algorithm 1 and thus to approximate the law of the target diffusion process $(X_t)_{t \ge 0}$.

In terms of Algorithm 1, we need to compute for three adjacent points $a, x, b$ of the grid the quantities

$$ v_0(x) = P_x(\tau_b < \tau_a), \quad v_1(x) = E_x(\tau_{ab} 1_{\tau_b < \tau_a}) \quad \text{and} \quad \bar{v}_1(x) = E_x(\tau_{ab} 1_{\tau_a < \tau_b}). $$

The quantities of (5.5) are then

$$ p^+[x, (a, b)] = v_0(x), \qquad T^+[x, (a, b)] = \frac{v_1(x)}{v_0(x)}, $$

$$ p^-[x, (a, b)] = 1 - v_0(x), \quad T^-[x, (a, b)] = \frac{\bar{v}_1(x)}{1 - v_0(x)}. $$

**Proposition 5.3.1.** The function $v_0(x) = P_x(\tau_b < \tau_a)$ is solution to the problem with Dirichlet boundary conditions

$$ \begin{cases} L u = 0, & x \in (a, b), \\ u(a) = 0, \\ u(b) = 1, \end{cases} \qquad (5.19) $$

where $L$ is the infinitesimal generator of $(X_t)_{t \ge 0}$, which also implies that $v_0 \in \mathrm{dom}(L)$.

*Proof.* Let $x \in (a, b)$, from the definition of the scale function and the factorization of the infinitesimal generator $L = D_m D_s$

$$ L v_0 = D_m D_s \frac{s(\cdot) - s(a)}{s(b) - s(a)} = D_m \frac{1}{s(b) - s(a)}. $$
which equals 0 as $m(dx)$ is a positive measure. As $v_0$ and $L v_0 = 0$ are both functions in $C_b^0$, we deduce that $v_0 \in \text{dom}(L)$ and $L v_0 = 0$. Under $P_b$, the stopping time $\tau_b$ equal 0 a.s. and the process has a.s. continuous trajectories, hence $\tau_a > 0$ a.s., i.e.,

$$v_0(b) = P_b(\tau_b < \tau_a) = P_b(0 < \tau_a) = 1.$$

This, along with the symmetrical argument, allow us to retrieve the boundary conditions of (5.19). □

**Proposition 5.3.2.** For every $k \in \mathbb{N}$, let $v_k$ be the function defined for every $x \in (a, b)$ by $v_k(x) = E_x(\tau_{ab}^k 1_{\tau_b < \tau_a})$. Then,

$$v_k(x) = k \int_{(a,b)} G_{a,b}(x, \zeta) v_{k-1}(\zeta) m(d\zeta). \qquad (5.20)$$

*Proof.* Since $\int_0^{\tau_{ab}} (\tau_{ab} - t)^{k-1} dt = \tau_{ab}^k/k$,

$$v_k(x) = k E_x \left[ 1_{\tau_b < \tau_a} \int_0^{\tau_{ab}} (\tau_{ab} - t)^{k-1} dt \right] = k E_x \left[ 1_{\tau_b < \tau_a} \int_0^{\infty} 1_{t \le \tau_{ab}} (\tau_{ab} - t)^{k-1} dt \right].$$

From the Markov property, by conditioning on $\mathcal{F}_t$ and as $1_{t \le \tau_{ab}}$ is $\mathcal{F}_t$-measurable,

$$v_k(x) = k E_x \left[ \int_0^\infty 1_{t \le \tau_{ab}} E \left[ 1_{\tau_b < \tau_a} (\tau_{ab} - t)^{k-1} \middle| \mathcal{F}_t \right] dt \right] = k E_x \left[ \int_0^{\tau_{ab}} E_{X_t} \left[ 1_{\tau_b < \tau_a} \tau_{ab}^{k-1} \right] dt \right].$$

The equality (5.20) results by applying directly Green's formula. □

**Proposition 5.3.3.** The function $v_k(x) = E_x(\tau_{ab}^k 1_{\tau_b < \tau_a})$ is solution to the problem with Dirichlet boundary conditions

$$\begin{cases} L u = -kv_{k-1}, & x \in (a, b), \\ u(a) = 0, \\ u(b) = 0. \end{cases}$$

**Lemma 5.3.4.** Let $g(x) = \int_{(a,b)} G_{a,b}(x, y)f(y)m(dy)$, where $f \in C_b^0(a, b)$ and $G_{a,b}(x, y)$ is the Green function defined in (1.20). Then $g \in \text{dom}(L)$ and

$$L g(x) = -f(x), \quad \forall x \in (a, b).$$

*Proof.* Let $x \in (a, b)$. Using the $D_m D_s$ factorization of $L$ and the dominated convergence theorem we get

$$\begin{align*} L g(x) &= D_m D_s \int_{(a,b)} \left[ 1_{y<x} \frac{(s(y)-s(a))(s(b)-s(x))}{s(b)-s(a)} + 1_{y \ge x} \frac{(s(x)-s(a))(s(b)-s(y))}{s(b)-s(a)} \right] f(y)m(dy) \\ &= -D_m \int_{(a,x)} v_0(y)f(y)m(dy) + D_m \int_{(x,b)} (1-v_0(y))f(y)m(dy) \\ &= -v_0(x)f(x) - (1-v_0(x))f(x) = -f(x). \end{align*}$$

The continuity of $g$ is a consequence of Lebesgue's convergence theorem for integrals. Moreover as $G_{a,b}(x, y)$ is bounded by $s(b) - s(a)$, $m(dx)$ is locally finite and $f$ is bounded, $g$ is also bounded. So, we deduce that $f \in \text{dom}(L)$ and that on $(a, b)$ we have $L g = -f$. □
Proof of Proposition 5.3.3. From Proposition 5.3.2,

$$v_k(x) = \int_{(a,b)} G_{a,b}(x, y)kv_{k-1}(y)m(dy).$$

As $v_0 \in C_b^0(a, b)$, from Lemma 5.3.4 we deduce iteratively that $v_k \in C_b^0(a, b)$ for all $k \in \mathbb{N}$ and that $L v_k = -kv_{k-1}$ on $(a, b)$. For the boundary conditions, we observe that $\tau_{ab} = \tau_a \wedge \tau_b = 0$ a.s. under $P_a$, so for $k \ge 1$

$$v_k(a) = E_a(\tau_{ab}^k \mathbb{1}_{\tau_b < \tau_a}) \le E_a(\tau_{ab}^k) = 0.$$

With the same argument we show that $v_k(b) \le 0$ and as they are obviously positive quantities $v_k(a) = v_k(b) = 0$. $\square$

## 5.4 Convergence of the embedding times

In order to prove the convergence of the process $(\tilde{X}_t^g)_{t \ge 0}$ we need to control quantities of the form $E_x[\sup_{t \le T} |t - \tau_{Kg(T)}^g|^p]$, where $\tau_k^g$ are the embedding times of the process $X$ in the grid $g$. In this section, we show the existence of such bounds in terms of the metric

$$|g|_X = \sup_{c \in \mathcal{C}(g)} \{s(c)m(c)\}$$

defined in (5.1). If there exists a constant $K > 0$ such that $m(c) \le K$ for all $c \in \mathcal{C}(g)$, then, $|c|m(c) \le K|c|$ and $|g|_X \le K|g|$. Thus, we can bound the quantities of interest in terms of $|g|$ instead of $|g|_X$. But, in doing so, we do not track correctly the convergence rates of the approximation process. For example, for the standard Brownian motion, $|g|_B = |g|^2$. Moreover, as shown in Section 5.2.3, such bounds give us a direct way to adapt the grid to the speed measure in order to accelerate the convergence of the scheme.

### 5.4.1 Bounds on the conditional moments of the exit times

**Lemma 5.4.1.** Let $v_k(x) = E_x(\tau_{ab}^k \mathbb{1}_{\tau_b < \tau_a})$, then for all $k \in \mathbb{N}$

$$\left\| \frac{v_k}{v_{k-1}} \right\|_{\infty} \le k|g|_X. \qquad (5.21)$$

*Proof.* We first observe that

$$G_{a,b}(x, y) \frac{v_0(y)}{v_0(x)} = \begin{cases} \frac{(y-a)(b-x)}{(b-a)} \frac{y-a}{x-a}, & x > y, \\ \frac{(x-a)(b-y)}{(b-a)} \frac{y-a}{x-a}, & x \le y. \end{cases}$$

As for $x > y$ the ratio $\frac{y-a}{x-a} < 1$,

$$G_{a,b}(x, y) \frac{v_0(y)}{v_0(x)} \le \begin{cases} \frac{(y-a)(b-x)}{(b-a)}, & x > y, \\ \frac{(y-a)(b-y)}{(b-a)}, & x \le y, \end{cases}$$
which is bounded by $(b-a)$ in both cases as $x, y \in (a, b)$. Thus,

$$ \int_a^b G_{a,b}(x, y)v_0(y)m(dy) = v_0(x) \int_a^b G_{a,b}(x, y) \frac{v_0(y)}{v_0(x)}m(dy) \le v_0(x)(b-a)m((a, b)). \quad (5.22) $$

From Proposition 5.3.2, $v_k(x) = k \int_{(a,b)} G_{a,b}(x, y)v_{k-1}(y)m(dy)$ and

$$ v_k(x) = k! \int_{(a,b)} G_{a,b}(x, x_k) \int_{(a,b)} G_{a,b}(x_k, x_{k-1}) \cdots \int_{(a,b)} G_{a,b}(x_2, x_1) v_0(x_1) m(dx_1) \cdots m(dx_k). \quad (5.23) $$

From (5.22) and (5.23),

$$ v_k(x) \le k(b-a)m((a,b))v_{k-1}(x). $$

Since $(b-a)m((a,b)) \le |\mathbf{g}|_X$, we get the desired result on $v_k/v_{k-1}$. $\square$

**Corollary 5.4.2.** Let $k \in \mathbb{N}$. Then, we have the following bound for $v_k(x) = E_x(\tau_{ab}^k 1_{\tau_b < \tau_a})$,

$$ \|v_k\|_{\infty} \le k! |\mathbf{g}|_X^k. $$

**Lemma 5.4.3.** Let $X$ be a diffusion process with state space $\mathbb{I}$ an interval of $\mathbb{R}$, $m$ the speed measure of $X$, $a, x, b \in \mathbb{I}$ such that $a < x < b$, $\mathbf{c}_x = (a, b)$, $|\mathbf{c}_x|_X = m(\mathbf{c}_x)|b-a|$ and $\lambda > 0$ such that $\lambda|\mathbf{c}_x|_X \in (0, 1)$. Then,

$$ E_x(e^{\lambda \tau_{ab}}) \le \exp \left( \frac{\lambda}{1 - \lambda |\mathbf{c}_x|_X} E_x(\tau_{ab}) \right). \quad (5.24) $$

*Proof.* Developing the exponential series,

$$ E_x(e^{\lambda \tau_{ab}}) = 1 + \sum_{N=1}^{\infty} \frac{\lambda^N}{N!} E_x(\tau_{ab}^N) = 1 + E_x(\tau_{ab}) \sum_{N=1}^{\infty} \frac{\lambda^N}{N!} \frac{E_x(\tau_{ab}^N)}{E_x(\tau_{ab})}. $$

From Corollary 5.4.2, we can bound the ratio of expected values by $(N-1)!|\mathbf{c}_x|_X^{N-1}$. As $\lambda|\mathbf{c}_x|_X \in (0, 1)$,

$$ E_x(e^{\lambda \tau_{ab}}) \le 1 + E_x(\tau_{ab}) \sum_{N=1}^{\infty} \frac{\lambda}{N} (\lambda |\mathbf{c}_x|_X)^{N-1} \le 1 + E_x(\tau_{ab}) \frac{\lambda}{1 - \lambda |\mathbf{c}_x|_X}. $$

Thus, we only need to apply the inequality $1+x \le e^x$ to get (5.24). $\square$

**Lemma 5.4.4.** Let $t, M > 0$ and $\lambda > 0$ such that $\lambda|\mathbf{g}|_X \in (0, 1)$. Then,

$$ P_x(\tau_{K\mathbf{g}(t)}^g > M) \le e^{-\lambda M} e^{\lambda \frac{t+|\mathbf{g}|_X}{1-\lambda|\mathbf{g}|_X}}. \quad (5.25) $$

*Proof.* From Markov's inequality,

$$ P_x(\tau_{K\mathbf{g}(t)}^g > M) \le e^{-\lambda M} E_x \left[ e^{\lambda \tau_{K\mathbf{g}(t)}^g} \right] = e^{-\lambda M} E_x \left[ e^{\lambda \sum_{k=1}^{K\mathbf{g}(t)} (\tau_k - \tau_{k-1})} \right]. $$

Conditioning on the $\sigma$-algebra $\mathcal{B}$ generated by the trajectory of $X_t$ on the grid $\mathbf{g}$, i.e., $\mathcal{B} = \sigma\{X_{\tau_k}; k \in \mathbb{N}_0\}$, and as $K\mathbf{g}(t)$ is $\mathcal{B}$-measurable,

$$ E_x \left[ e^{\lambda \sum_{k=1}^{K\mathbf{g}(t)} (\tau_k - \tau_{k-1})} \right] = E_x \left[ \prod_{k=1}^{K\mathbf{g}(t)} E \left[ e^{\lambda (\tau_k - \tau_{k-1})} \middle| \mathcal{B} \right] \right]. \quad (5.26) $$
From the definition of $|g|_X$, $\lambda|c|_X \le \lambda|g|_X \in (0, 1)$ for each cell $c$ of the grid $g$. Thus, applying Lemma 5.4.3 on each term of the product in (5.26),

$$ P_x(\tau_{Kg(t)}^g > M) \le e^{-\lambda M} E_x \left[ \exp \left( \frac{\lambda}{1 - \lambda|g|_X} \sum_{k=1}^{Kg(t)} E(\tau_k - \tau_{k-1}|\mathcal{B}) \right) \right] \quad (5.27) $$

From (5.18),

$$ t < \sum_{k=1}^{Kg(t)} E(\tau_k - \tau_{k-1}|\mathcal{B}) \le t + E(\tau_{Kg(t)+1} - \tau_{Kg(t)}|\mathcal{B}) $$

Thus, from (5.21),

$$ \sum_{k=1}^{Kg(t)} E(\tau_k - \tau_{k-1}|\mathcal{B}) \le t + \|v_1/v_0\|_{\infty} \le t + |g|_X, \quad (5.28) $$

From (5.27) and (5.28), we get (5.25). $\square$

## 5.4.2 Convergence of the embedding times

For this section, let $X$ be a diffusion process with state-space $\mathbb{I}$ an interval of $\mathbb{R}$, $g$ a covering grid of $\mathbb{I}$, $\tau_k^g$ the embedding times of $X$ in $g$ as defined in (5.17) and $K^g(t)$ as defined in (5.18).

**Lemma 5.4.5.** For any $T > 0$,

$$ \sum_{k=1}^{Kg(T)} \mathrm{Var} \left( \tau_k^g - \tau_{k-1}^g \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right) \le 2|g|_X (T + |g|_X), $$

where $|\cdot|_X$ is defined in (5.1).

*Proof.* For all $x \in g$, let $c_x$ be the cell of $g$ containing $x$. Then,

$$ \begin{aligned} \sum_{k=1}^{Kg(T)} & \mathrm{Var} \left( \tau_k^g - \tau_{k-1}^g \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right) \\ &= \sum_{k=1}^{Kg(T)} E \left[ (\tau_k^g - \tau_{k-1}^g)^2 \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] - \left( E \left[ \tau_k^g - \tau_{k-1}^g \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] \right)^2 \\ &\le \sup_{k \le Kg(t)} \left\{ \frac{E \left[ (\tau_k^g - \tau_{k-1}^g)^2 \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right]}{E \left[ \tau_k^g - \tau_{k-1}^g \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right]} \right\} \sum_{k=1}^{Kg(T)} E \left[ (\tau_k^g - \tau_{k-1}^g) \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right]. \end{aligned} $$

So from Lemma 5.4.1 and the definition of $K^g(t)$,

$$ \begin{aligned} \sum_{k=1}^{Kg(T)} & \mathrm{Var} \left( \tau_k^g - \tau_{k-1}^g \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right) \\ & \le \left\| \frac{v_2}{v_1} \right\|_{\infty} \sum_{k=1}^{Kg(T)} E \left[ (\tau_k^g - \tau_{k-1}^g) \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] \\ & \le \left\| \frac{v_2}{v_1} \right\|_{\infty} \left( T + \left\| \frac{v_1}{v_0} \right\|_{\infty} \right) \le 2|g|_X(T + |g|_X), \end{aligned} $$

which is the desired inequality. $\square$
**Proposition 5.4.6.** Let $(\mathcal{F}_t)_{t \ge 0}$ be the canonical filtration of X. Let also $\mathcal{A}_n = \mathcal{F}_{\tau_n^g}$, $\mathcal{B} = \sigma((X_{\tau_k^g})_{k \in \mathbb{N}_0})$ and $\Delta \tau_k^g = \tau_k^g - \tau_{k-1}^g$. If we define the augmented filtration $\mathcal{G}_n = \mathcal{A}_n \vee \mathcal{B}$, then the process

$$M_n = \sum_{k=1}^{n} \Delta \tau_k^g - E \left[ \Delta \tau_k^g \middle| X_{\tau_{k-1}^g}, X_{\tau_k^g} \right],$$

is a $\mathcal{G}_n$-martingale.

*Proof.* For $m \le n$,

$$\begin{aligned} E[M_n | \mathcal{G}_m] &= E[M_n | \mathcal{A}_m, \mathcal{B}] \\ &= E \left[ \sum_{k=1}^{m} \Delta \tau_k^g - E \left[ \Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] + \sum_{k=m+1}^{n} \Delta \tau_k^g - E \left[ \Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] \middle| \mathcal{A}_m, \mathcal{B} \right]. \end{aligned}$$

From Lemma A.4.1 and as $\tau_k^g$ is $\mathcal{A}_k$ measurable,

$$E[M_n | \mathcal{G}_m] = M_m + \sum_{k=m+1}^{n} E_{X_{\tau_k^g}} \left[ \Delta \tau_k^g - E \left[ \Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] | \mathcal{B} \right] = M_m.$$

This proves the result.

<figure><img src="image_14.png" /></figure>

**Theorem 5.4.7.** For any $T > 0$,

$$E \left[ \sup_{t \in [0,T]} |\tau_{Kg(t)}^g - t|^2 \right] \le 2|g|_X \left( 4(T + |g|_X) + 1 \right), \quad (5.29)$$

where $|\cdot|_X$ is defined in (5.1).

*Proof.* The convexity inequality $(a+b)^p \le 2^{p-1}(a^p + b^p)$ yields for $p = 2$

$$\begin{aligned} E \left[ \sup_{t \in [0,T]} |\tau_{Kg(t)}^g - t|^2 \right] &\le 2 E \left[ \sup_{t \in [0,T]} \left| \tau_{Kg(t)}^g - \sum_{k=1}^{Kg(t)} E \left[ \Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] \right|^2 \right] \\ &\quad + 2 E \left[ \sup_{t \in [0,T]} \left| \sum_{k=1}^{Kg(t)} E \left[ \Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] - t \right|^2 \right]. \end{aligned} \quad (5.30)$$

From Proposition 5.4.6, the term $\sum_{k=1}^n \Delta \tau_k^g - E[\Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g}]$ is a $\mathcal{G}_n$-martingale, where $\mathcal{G}_n = \mathcal{F}_{\tau_n^g} \vee \mathcal{B}$ and $\mathcal{B} = \sigma((X_{\tau_k^g})_{k \in \mathbb{N}_0})$. Thus, from Doob's $L^p$ inequality,

$$E \left[ \sup_{t \in [0,T]} |M_{Kg(t)}|^2 \right] = E \left[ \sup_{k \le Kg(T)} |M_k|^2 \right] \le 2 E \left[ |M_{Kg(T)}|^2 \right].$$

By conditioning on $\mathcal{B}$, from Lemma A.4.1,

$$\begin{aligned} &E \left[ \sup_{t \in [0,T]} \left| \sum_{k=1}^{Kg(t)} \Delta \tau_k^g - E \left[ \Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] \right|^2 \right] \\ &\le 2 E \left[ \left( \sum_{k=1}^{Kg(T)} \Delta \tau_k^g - E \left[ \Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g} \right] \right)^2 \right] \\ &= 2 E \left[ \sum_{k=1}^{Kg(T)} \mathrm{Var} \left( \Delta \tau_k^g | X_{\tau_{k-1}^g}, X_{\tau_k^g} \right) \right], \end{aligned}$$
which from Lemma 5.4.5 is bounded by $4|g|_X(T + |g|_X)$. For the second term on the right hand side of (5.30), from (5.18) and since $K^g(t) \ge 1$ for any $t > 0$,

$$ \sum_{k=1}^{K^g(t)} E[\tau_k^g - \tau_{k-1}^g | X_{\tau_{k-1}^g}, X_{\tau_k^g}] - t \le E[\tau_{K^g(t)}^g - \tau_{K^g(t)-1}^g | X_{\tau_{K^g(t)-1}^g}, X_{\tau_{K^g(t)}}^g] \\ \le \|\frac{v_1}{v_0}\|_{\infty} \le |g|_X. $$

So having bounded both additive parts of the right hand side of (5.30), we get (5.29). □

# 5.5 Convergence rate of the Markov chain

## 5.5.1 Moment bounds

In this section we prove that Theorem 3.1 of [9] holds also for reflected processes and for a sharper constant. This result, combined with the bound (5.29) is used to prove the convergence of the approximation process in Section 5.5.2.

**Lemma 5.5.1.** Let $X$ be a diffusion process on natural scale with state-space $\mathbb{R}$ and a speed measure $m_X$ that satisfies Condition (5.7). Then, for all $p \ge 2$, there exist two constants $C, C' > 0$ such that

$$ E_x \left[ \sup_{t \in [0,T]} |X_t - x|^p \right] \le C' [1 + |x|^p] e^{CT}, \quad (5.31) $$

where $k_1$ is a constant such that Condition (5.7) is satisfied, $C \le 8p(p-1)/k_1$ and $C' > 0$ is a constant that depends only on $p$.

*Proof.* Let $Z$ be the diffusion process on natural scale with speed measure,

$$ m_Z(dx) = 1_{|x|<1} \frac{k_1}{2} dx + 1_{|x| \ge 1} \frac{k_1}{2x^2} dx. $$

We note that $\frac{k_1}{2x^2} \le \frac{k_1}{1+x^2}$ for all $x \ge 1$. The dynamic of $Z$ can be shown to be

$$ dZ_t = \begin{cases} \frac{2}{\sqrt{k_1}} dB_t, & \text{for } |Z_t| < 1, \\ \frac{2}{\sqrt{k_1}} Z_t dB_t, & \text{for } |Z_t| \ge 1, \end{cases} $$

where $B$ is a standard Brownian motion. We also assume that $P_x(Z_0 = x) = 1$. As $X$ and $Z$ are on natural scale, they can be expressed as time-changed Brownian motion [82, Theorem 47.1, p. 277], i.e., for every $t \ge 0$, $X_t = B_{\gamma_X(t)}$ and $Z_t = W_{\gamma_Z(t)}$, where $B$ and $W$ are two standard Brownian motions with $P_x(B_0 = x) = P_x(W_0 = x) = 1$, respective local times $L^x(B), L^x(W)$ and with $\gamma_X(t), \gamma_Z(t)$ being the respective right-inverses<fnref n="6" /> of

$$ A_X(t) = \frac{1}{2} \int_{\mathbb{I}} L_t^x(B) m_X(dx) \quad \text{and} \quad A_Z(t) = \frac{1}{2} \int_{\mathbb{I}} L_t^x(W) m_Z(dx). $$

$$ f^{-1}(x) = \inf\{\zeta \ge 0 : f(\zeta) > x\}. $$

<footnote n="6">⁶The right-inverse of a function $f$ is given by,</footnote>
Using the same underlying Brownian motion in these definitions, we have $A_Z(t) \le A_X(t)$, and hence $\gamma_X(t) \le \gamma_Z(t)$. Thus,

$$ 
\begin{aligned}
E_x \left[ \sup_{t \in [0,T]} |X_t - x|^p \right] &= E_x \left[ \sup_{t \in [0,\gamma_X(T)]} |B_t - x|^p \right] \\
&\le E_x \left[ \sup_{t \in [0,\gamma_Z(T)]} |B_t - x|^p \right] = E_x \left[ \sup_{t \in [0,T]} |Z_t - x|^p \right].
\end{aligned}
$$

Thus, from Doob's $L^p$ and convexity inequalities,

$$ E_x \left[ \sup_{t \in [0,T]} |X_t - x|^p \right] \le \frac{2^{p-1}p}{p-1} \left( E_x \left[ |Z_T|^p \right] + |x|^p \right). \quad (5.32) $$

Using Itô's formula, followed by a classical localization argument, Fatou's Lemma along with the standard dominated convergence theorem, one obtains that for all $q > 2/3$,

$$ 
\begin{aligned}
E_x \left[ (|Z_t| - 1)^{3q} 1_{|Z_t| \ge 1} + 1 \right] &\le (|x| - 1)^{3q} 1_{|x| \ge 1} + 1 + \int_0^t \frac{3q}{2} (3q - 1) E_x \left[ (|Z_s| - 1)^{3q-2} 1_{|Z_s| \ge 1} \frac{4Z_s^2}{k_1} \right] ds.
\end{aligned}
$$

Using the inequality $|x - 1|^{3q-2}x^2 \le 4|x - 1|^{3q} + 4$ for all $x \ge 1$,

$$ 
\begin{aligned}
E_x \left[ (|Z_t| - 1)^{3q} 1_{|Z_t| \ge 1} + 1 \right] &\le (|x| - 1)^{3q} 1_{|x| \ge 1} + 1 + \int_0^t \frac{24q}{k_1} (3q - 1) E_x \left[ (|Z_s| - 1)^{3q} 1_{|Z_s| \ge 1} + 1 \right] ds.
\end{aligned}
$$

Using Gronwall's Lemma, we deduce that, for all $t \ge 0$,

$$ E_x \left[ (|Z_t| - 1)^{3q} 1_{|Z_t| \ge 1} + 1 \right] \le \left[ (|x| - 1)^{3q} 1_{|x| \ge 1} + 1 \right] e^{24q(3q-1)t/k_1}. $$

Let $C'_q > 0$ be a constant such that $(|x| - 1)^{3q} 1_{|x| \ge 1} + 1 \ge C'_q |x|^{3q}$ for all $x \in \mathbb{R}$. Thus,

$$ E_x \left[ |Z_t|^{3q} \right] \le C'_q \left[ (|x| - 1)^{3q} 1_{|x| \ge 1} + 1 \right] e^{24q(3q-1)t/k_1}, \quad \forall t \ge 0. $$

Hence, for all $p > 2$, we have

$$ E_x \left[ |Z_t|^p \right] \le C'_{p/3} \left[ 1 + |x|^p \right] e^{8p(p-1)t/k_1}, \quad \forall t \ge 0. \quad (5.33) $$

Replacing (5.33) in (5.32), we get the bound (5.31) for $C' = \frac{p}{p-1} 2^{p-1}(C'_{p/3} + 1)$ and $C = 8p(p-1)/k_1$.

□

**Proposition 5.5.2.** Let $X$ be a diffusion process with state-space $\mathbb{I}$ an interval of $\mathbb{R}$, on natural scale and with a speed measure $m_X$ that satisfies Condition (5.7). Then, for each $T > 0$ and $\gamma \in (0, \frac{1}{2})$, there exists a constant $C > 0$ such that

$$ \left\| \sup_{s \neq t \leq T} \frac{|X_t - X_s|}{|t - s|^{\gamma}} \right\|_{L^p(\mathbb{P}_x)} \leq C(1 + |x|). \quad (5.34) $$

The result also holds in the presence of a reflecting boundary $\zeta \in \mathbb{I}$.

<footnote n="7">⁷The extension to $p = 2$ is straightforward.</footnote>
*Proof.* For the non-reflecting case, the proof works using the same arguments as in the proof of Theorem 3.1 in [9].

For the reflecting case: Let $X$ be a diffusion process on natural scale, with speed measure $m$ satisfying (5.9) for a constant $k_1 > 0$ and a reflecting boundary at $\zeta \in \mathbb{I}$. We observe that

$$X = |X^{\circ} - \zeta| + \zeta \qquad (5.35)$$

in law, where $X^{\circ}$ is the non-reflecting diffusion on natural scale with speed measure

$$m_{X^{\circ}}(dx) = 1_{x \ge \zeta} m(dx) + 1_{x < \zeta} m(2\zeta - dx)$$

which also satisfies (5.9) for the same constant $k_1$. From (5.35) and the triangle inequality,

$$\frac{|X_t - X_s|}{|t - s|^{\gamma}} = \frac{|X_t^{\circ} - \zeta| - |X_s^{\circ} - \zeta|}{|t - s|^{\gamma}} \le \frac{|X_t^{\circ} - X_s^{\circ}|}{|t - s|^{\gamma}}. \qquad (5.36)$$

The diffusion process $X^{\circ}$ is non-reflecting and its speed measure $m_{X^{\circ}}$ satisfies (5.9). Thus, (5.34) holds for $X^{\circ}$ for a constant $C > 0$ and from (5.36),

$$\sup_{s \neq t \leq T} \frac{|X_t - X_s|}{|t - s|^{\gamma}} \leq \sup_{s \neq t \leq T} \frac{|X_t^{\circ} - X_s^{\circ}|}{|t - s|^{\gamma}} \leq C(1 + |x|). \quad \square$$

### 5.5.2 Proof of the convergence rate for a process on natural scale

*Proof of Theorem 5.2.1.* From Proposition 5.2.7,

$$\begin{aligned} \mathcal{W}_p \left[ \text{Law} \left( (\widetilde{X}_t^g)_{t \in [0,T]} \right), \text{Law} \left( (X_t)_{t \in [0,T]} \right) \right] \\ = \inf \left\{ \begin{array}{l} \|\zeta - \xi\|_{L^p} \\ \quad ; \zeta \sim \text{Law} \left( (\widetilde{X}_t^g)_{t \in [0,T]} \right), \xi \sim \text{Law} \left( (X_t)_{t \in [0,T]} \right) \end{array} \right\} \\ \le \left\| \sup_{t \in [0,T]} |X_{\tau_{Kg(t)}^g} - X_t| \right\|_{L^p(P_x)}. \end{aligned}$$

Let $M > T$. From Minkowski inequality,

$$\left\| \sup_{t \in [0,T]} |X_{\tau_{Kg(t)}^g} - X_t| \right\|_{L^p(P_x)} \le \left\| \mathbb{1}_{\tau_{Kg(t)}^g \le M} \sup_{t \in [0,T]} |X_{\tau_{Kg(t)}^g} - X_t| \right\|_{L^p(P_x)} \\ + \left\| \mathbb{1}_{\tau_{Kg(t)}^g > M} \sup_{t \in [0,T]} |X_{\tau_{Kg(t)}^g} - X_t| \right\|_{L^p(P_x)} \quad (5.37)$$

For the first additive term of (5.37), for any $\gamma > 0$, by multiplying and dividing by $|\tau_{Kg(t)}^g - t|^{\gamma}$,

$$\begin{aligned} \left\| \mathbb{1}_{\tau_{Kg(t)}^g \le M} \sup_{t \in [0,T]} |X_{\tau_{Kg(t)}^g} - X_t| \right\|_{L^p(P_x)} \\ = \left\| \mathbb{1}_{\tau_{Kg(t)}^g \le M} \sup_{t \in [0,T]} \left\{ \frac{|X_{\tau_{Kg(t)}^g} - X_t|}{|\tau_{Kg(t)}^g - t|^{\gamma}} \right\} \tau_{Kg(t)}^g - t \right\|_{L^p(P_x)} \\ \le \left\| \mathbb{1}_{\tau_{Kg(t)}^g \le M} \sup_{t \in [0,T]} \left\{ \frac{|X_{\tau_{Kg(t)}^g} - X_t|}{|\tau_{Kg(t)}^g - t|^{\gamma}} \right\} \tau_{Kg(t)}^g - t \right\|_{L^p(P_x)}. \end{aligned}$$
As $\tau_{Kg(t)}^g$ is increasing with respect to $t$,

$$ \begin{aligned} \left\| \mathbb{1}_{\tau_{Kg}(T)} \le M \sup_{t \in [0,T]} \left| X_{\tau_{Kg}(t)} - X_t \right| \right\|_{L^p(\mathbb{P}_x)} & \le \left\| \sup_{s \ne t \le M} \left\{ \frac{|X_s - X_t|}{|s-t|^\gamma} \right\} \sup_{t \in [0,T]} |\tau_{Kg}(t) - t|^\gamma \right\|_{L^p(\mathbb{P}_x)} \\ & = \left( \mathrm{E}_x \left[ \sup_{s \ne t \le M} \left\{ \frac{|X_s - X_t|}{|s-t|^\gamma} \right\} \sup_{t \in [0,T]} |\tau_{Kg}(t) - t|^\gamma \right] \right)^{1/p}. \end{aligned} $$

From Hölder's inequality for $q \ge 1$ and $q/(q-1)$ conjugates exponents and (5.34), for every $\gamma \in (0, \frac{1}{2})$, there exists a constant $C_1 = C_1(M, \gamma, p(q-1)/q, x) > 0$ such that

$$ \begin{aligned} \left\| \mathbb{1}_{\tau_{Kg}(T)} \le M \sup_{t \in [0,T]} \left| X_{\tau_{Kg}(t)} - X_t \right| \right\|_{L^p(\mathbb{P}_x)} & \le \left( \mathrm{E}_x \left[ \sup_{s \ne t \le M} \frac{|X_s - X_t|}{|s-t|^\gamma} \right]^{p(q-1)/q} \right)^{q/p(q-1)} \\ & \qquad \left( \mathrm{E}_x \left[ \sup_{t \in [0,T]} |\tau_{Kg}(t) - t|^\gamma \right]^{pq} \right)^{1/pq} \\ & \le C_1^{q/p(q-1)} \left( \mathrm{E}_x \left[ \sup_{t \in [0,T]} |\tau_{Kg}(t) - t|^{\gamma pq} \right] \right)^{1/pq}. \end{aligned} $$

Choosing $q = 2/\gamma p$, from (5.29), for any $\gamma \in (0, \frac{1}{2} \wedge \frac{2}{p})$

$$ \left\| \mathbb{1}_{\tau_{Kg}(T)} \le M \sup_{t \in [0,T]} \left| X_{\tau_{Kg}(t)} - X_t \right| \right\|_{L^p(\mathbb{P}_x)} \le C_1^{1/p(1-\gamma p)} (2|g|_X (4T + |g|_X))^{\gamma/2}. \quad (5.38) $$

For the second additive term of (5.37),

$$ \mathrm{E}_x \left[ \mathbb{1}_{\tau_{Kg}(T)} > M \sup_{t \in [0,T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] = \sum_{m=M}^{\infty} \mathrm{E}_x \left[ \mathbb{1}_{\tau_{Kg}(T) \in [m,m+1]} \sup_{t \in [0,T]} |X_{\tau_{Kg}(t)} - X_t|^p \right]. \quad (5.39) $$

For each term of the sum in (5.39), from Hölder's inequality,

$$ \begin{aligned} & \mathrm{E}_x \left[ \mathbb{1}_{\tau_{Kg}(T) \in [m,m+1]} \sup_{t \in [0,T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] \\ & \le \left[ \mathrm{P}_x (\tau_{Kg}(T) > m) \right]^{1/q'} \left[ \mathrm{E}_x \left[ \mathbb{1}_{\tau_{Kg}(T) \in [m,m+1]} \sup_{t \in [0,T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] \right]^{q'/(q'-1)} \\ & \le \left[ \mathrm{P}_x (\tau_{Kg}(T) > m) \right]^{1/q'} \left[ \mathrm{E}_x \left[ \mathbb{1}_{\tau_{Kg}(T) < m+1} \sup_{t \in [0,T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] \right]^{q'/(q'-1)}. \end{aligned} $$

As each $m$ in the sum in (5.39) satisfies $m+1 > M > T$, from Minkowski's inequality,

$$ \begin{aligned} & \mathrm{E}_x \left[ \mathbb{1}_{\tau_{Kg}(T) < m+1} \sup_{t \in [0,T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] \\ & \le \mathrm{E}_x \left[ \mathbb{1}_{\tau_{Kg}(T) < m+1} 2^{p-1} \sup_{t \in [0,T]} \left\{ |X_{\tau_{Kg}(t)} - x|^p + |X_t - x|^p \right\} \right] \\ & \le 2^p \mathrm{E}_x \left[ \sup_{t \in [0,m+1]} |X_t - x|^p \right], \end{aligned} $$
which from Lemma 5.5.1 is bounded by $C_2[1 + |x|^p]e^{C_3(m+1)}$, where $C_2$ and $C_3$ are positive constants depending on $p$. This, along with Lemma 5.4.4 and Hölder's inequality gives us for $\lambda > 0$ chosen such that $\alpha = \lambda|g|_X < 1$,

$$E_x \left[ \mathbb{1}_{\tau_{Kg}(T) \in [m, m+1)} \sup_{t \in [0, T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] \le \left[ C_2 [1 + |x|^{pq'}] e^{C_3(m+1)} \right]^{1/q'} \left[ e^{-\lambda m} e^{\lambda \frac{T+|g|_X}{1-\lambda|g|_X}} \right]^{(q'-1)/q'}$$

where $C_2(pq') \le 2pq'(pq' - 1)/c$ and $C_2(pq') > 0$ a positive constant depending only on $pq'$. Thus, setting $C_4(pq') := [C_2(pq') [1 + |x|^{pq'}]]^{1/q'} e^{C_3/q'} > 0$,

$$\begin{aligned} E_x \left[ \mathbb{1}_{\tau_{Kg}(T) \in [m, m+1)} \sup_{t \in [0, T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] & \le C_4 \exp \left\{ \frac{1}{q'} \left[ C_3 m + (q' - 1) \left( \lambda \frac{T + |g|_X}{1 - \lambda |g|_X} - \lambda m \right) \right] \right\} \\ & = C_4 \exp \left\{ \frac{1}{q'} \left[ C_3 m + (q' - 1) \frac{\alpha}{|g|_X} \left( \frac{T + |g|_X}{1 - \alpha} - m \right) \right] \right\} \\ & = C_5 \exp \left\{ \frac{1}{q'} \left[ C_3 m + (q' - 1) \frac{\alpha}{|g|_X} \left( \frac{T}{1 - \alpha} - m \right) \right] \right\}, \end{aligned}$$

where $C_5 := C_4 \exp(\frac{q'-1}{q'} \frac{\alpha}{1-\alpha})$. If we choose $q' > 1$ such that $^8 A := C_3 - (q'-1)\alpha/|g|_X < 0$,

$$\begin{aligned} \sum_{m=M}^{\infty} E_x \left[ \mathbb{1}_{\tau_{Kg}(T) \in [m, m+1]} \sup_{t \in [0, T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] & \le C_5 \exp \left\{ \frac{(q'-1)\alpha T}{(1-\alpha)|g|_X} \right\} \sum_{m=M}^{\infty} (e^A)^m = C_5 \exp \left\{ \frac{(q'-1)\alpha T}{(1-\alpha)|g|_X} \right\} \frac{e^{AM}}{1-e^A} \\ & = C_5 \exp \left\{ \frac{(q'-1)\alpha T}{(1-\alpha)|g|_X} \right\} \frac{\exp\left\{\left(C_3 - \frac{(q'-1)\alpha}{|g|_X}\right)M\right\}}{1 - \exp\left\{C_3 - \frac{(q'-1)\alpha}{|g|_X}\right\}}. \end{aligned}$$

If $M$ is chosen such that $M > T/(1-\alpha)$, then

$$\sum_{m=M}^{\infty} E_x \left[ \mathbb{1}_{\tau_{Kg}(T) \in [m, m+1]} \sup_{t \in [0, T]} |X_{\tau_{Kg}(t)} - X_t|^p \right] \le \frac{C_5 e^{C_3 M}}{1 - e^A} \exp \left\{ \frac{(q'-1)\alpha}{|g|_X} \left( \frac{T}{1-\alpha} - M \right) \right\}, \quad (5.40)$$

where the bound is $O(e^{-1/|g|_X})$, and can be rewritten as $C^{(1)}e^{-C^{(2)}/|g|_X}$. From (5.37), (5.38) and (5.40),

$$\left\| \sup_{t \in [0, T]} |X_{\tau_{Kg}(t)} - X_t| \right\|_{L^p(P_x)} \le C_1^{1/p(1-\gamma p)} (2|g|_X (4T + |g|_X))^{\gamma/2} + C^{(1)} e^{-C^{(2)}/|g|_X}.$$

As $|g|_X^\gamma$ and $e^{-1/|g|_X}$ are both $O(|g|_X^{\gamma/2})$, there exists a constant $C > 0$, such that,

$$\left\| \sup_{t \in [0, T]} |X_{\tau_{Kg}(t)} - X_t| \right\|_{L^p(P_x)} \le C |g|_X^{\gamma/2},$$

which is (5.8).

<figure><img src="image_15.png" /></figure>

<footnote>$^8$We remark that if this is satisfied for a grid $g$, then it is satisfied for all grids $g'$ such that $|g'|_X \le |g|_X$.</footnote>
## 5.6 Computations for the classical SDE case and beyond

In order to implement Algorithm 1 for simulating paths of a diffusion $X$, one needs two things. First, a grid $g$ adapted to the scale function and speed measure of $X$. Second, good approximations of the transition probabilities and conditional transition times of $X$ over $g$. The first point was covered in Section 5.2.3.

In this section, we show how to compute the quantities (5.5) in the pure SDE case. The extension to SDE solutions with point-wise singularities is straightforward. This allows us via Algorithm 1 to simulate all such processes.

Let $(\mu, \sigma)$ two real-valued functions that satisfy Condition 5.2.6 and $X$ be the diffusion that solves

$$dX_t = \mu(X_t) dt + \sigma(X_t) dB_t,$$

where $B$ is a standard Brownian motion. Let also $\mathbb{I}$ be the state-space of $X$. A straightforward computation using Itô's formula gives us the infinitesimal generator of $X$,

$$(L, \mathrm{dom}(L)) = \begin{cases} L f(x) = \mu(x)f'(x) + \frac{1}{2}\sigma^2(x)f''(x), & \forall f \in \mathrm{dom}(L), \\ \mathrm{dom}(L) = (f \in C_b(\mathbb{I}) : L f \in C(\mathbb{I})). \end{cases} \quad (5.41)$$

In particular, if $\mu$ and $\sigma$ are continuous, then, $\mathrm{dom}(L) = C^2(\mathbb{I})$. From Proposition 5.3.1 and (5.41) we can infer that,

$$s'(x) = e^{-\int_a^y \frac{2\mu(\zeta)}{\sigma^2(\zeta)} d\zeta} \quad \text{and} \quad m(dx) = \frac{1}{s'(x)} \frac{2}{\sigma^2(x)} dx. \quad (5.42)$$

Thus, from Proposition 5.3.1,

$$v_0(x) = \frac{\int_a^x e^{-\int_a^y \frac{2\mu(\zeta)}{\sigma^2(\zeta)} d\zeta} d\zeta}{\int_a^b e^{-\int_a^y \frac{2\mu(\zeta)}{\sigma^2(\zeta)} d\zeta} d\zeta}. \quad (5.43)$$

From (5.20) and (5.42),

$$\begin{aligned} v_1(x) &= \int_a^b G_{a,b}(x, \zeta) v_0(\zeta) \frac{1}{s'(x)} \frac{2}{\sigma^2(x)} d\zeta, \\ \bar{v}_1(x) &= \int_a^b G_{a,b}(x, \zeta) (1 - v_0(\zeta)) \frac{1}{s'(x)} \frac{2}{\sigma^2(x)} d\zeta. \end{aligned} \quad (5.44)$$

The scale functions and the speed measures are defined up to a multiplicative constant: for $\alpha \in \mathbb{R}$ and $\lambda > 0$, the pairs $(s, m)$ and $(\alpha + \lambda s, \lambda^{-1}m)$ are associated to the same diffusion. In particular, as we are only concerned with points $x \in [a, b]$, we could use $v_0$ for the scale function. The speed measure shall be adapted accordingly. For $x, \zeta \in [a, b]$, the Green function in (1.20) takes the simpler form,

$$G_{a,b}(x, \zeta) = v_0(x \wedge \zeta) (1 - v_0(x \vee \zeta)).$$
Thus, expressions (5.44) become

$$ 
\begin{aligned}
v_1(x) &= \int_a^b v_0(x \wedge \zeta) (1 - v_0(x \vee \zeta)) \frac{v_0(\zeta)}{v'_0(\zeta) \sigma^2(\zeta)} \frac{2}{\sigma^2(\zeta)} d\zeta, \\
\bar{v}_1(x) &= \int_a^b v_0(x \wedge \zeta) (1 - v_0(x \vee \zeta)) \frac{1 - v_0(\zeta)}{v'_0(\zeta) \sigma^2(\zeta)} \frac{2}{\sigma^2(\zeta)} d\zeta.
\end{aligned} 
\quad (5.45) 
$$

The quantities (5.43) and (5.44) or (5.45) can be computed analytically as we do for the Ornstein-Uhlenbeck process in Section 6.1.4 or approximated numerically as for the Cox-Ingersoll-Ross process in Section 6.1.4.
# Chapter 6

## Numerical experiments

**Chapter outline:** In this chapter we aim to test numerically the theoretical results established in this thesis, namely the STMCA approximation of one-dimensional diffusions of Chapter 5, the local time approximation and the stickiness parameter estimation of Chapter 4. In Section 6.1, we give numerical STMCA approximation examples of diffusions that exhibit various path-wise features. In Section 6.2, we assess numerically the stickiness parameter estimation and the local time approximation. We also see how the flexibility of grid choice in Algorithm 1 make it suitable for simulations regarding the local time approximation.

## 6.1 Approximation in distribution

### 6.1.1 Standard Brownian motion:

The standard Brownian motion can be defined as the diffusion process with scale function and speed measure

$$s(x) = x \quad \text{and} \quad m(dx) = 2dx.$$

Let $v_k(x) := E_x[\tau_{ab}^k 1_{\tau_b < \tau_a}]$ and $\bar{v}_k(x) := E_x[\tau_{ab}^k 1_{\tau_a < \tau_b}]$ for all $k \in \mathbb{N}_0$, where $c_x = (a, b)$. Then, from the definition of the scale function,

$$v_0(x) = \frac{x-a}{b-a}.$$

From Proposition 5.3.2,

$$v_1^{BM}(x) = (x-a)(b-x)\left(\frac{2}{3}\frac{(x-a)^2}{(b-a)^2} + \frac{b-x}{b-a} - \frac{2}{3}\frac{(b-x)^2}{(b-a)^2}\right),$$

$$\bar{v}_1^{BM}(x) = (x-a)(b-x)\left(\frac{2}{3}\frac{(b-x)^2}{(b-a)^2} + \frac{x-a}{b-a} - \frac{2}{3}\frac{(x-a)^2}{(b-a)^2}\right),$$

where by BM we mean these quantities are associated with the Brownian motion. Thus from (5.5), we have all the necessary quantities we need to implement the algorithm.
<figure><img src="image_16.png" /><figcaption>Figure 6.1: (a): histogram of simulated values at $T = 1$ of a sticky Brownian motion of parameter $\rho = 0.7$ with initial value $x_0 = 0$ using Algorithm 1 with the tuned grid (5.12) of size-criteria $h = 0.01$.</figcaption></figure>

(b): histogram of simulated values at $T = 1$ of a skew Brownian motion of parameter $\beta = 0.9$ with initial value $x_0 = 0$ using Algorithm 1 with a uniform grid of step-size $h = 0.01$.

<figure><img src="image_17.png" /><figcaption>Figure 6.2: Histogram of simulated values at $T = 1$ of a CIR process of $(\theta, \mu, \sigma) = (5, 5, 1)$ with initial value $x_0 = 1$ using Algorithm 1 with:</figcaption></figure>

(a): a uniform grid of step-size $h = 0.01$ and (250, 200)-step Riemann approximation of $(v_0, v_1)$ (simulation time: 44.5 sec).

(b): a tuned grid of size-criteria $h = 0.01$ computed solving numerically (5.15)-(5.16) with Newton's method and (250, 200)-step Riemann approximation of $(v_0, v_1)$ (simulation time: 46.5 sec).
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>Figure 6.3: Histogram of simulated values at $T = 1$ of the CIR process of $(\theta, \mu, \sigma) = (1, 2, 1)$ with initial value $x_0 = 5$ using Algorithm 1 with:</figcaption></figure>

(a): a uniform grid of step-size $h = 0.015$ and (100, 100)-step Riemann approximation of $(v_0, v_1)$ (simulation time: 10.8 sec).

(b): a tuned grid of size-criteria $h = 0.015$ computed solving numerically (5.15)-(5.16) with Newton's method and (100, 100)-step Riemann approximation of $(v_0, v_1)$ (simulation time: 12.8 sec).

(c): Same as Figure 6.3a but with a (250, 200)-step Riemann approximation of $(v_0, v_1)$ (simulation time: 11.5 sec).

(d): Same as Figure 6.3b but with a (250, 200)-step Riemann approximation of $(v_0, v_1)$ (simulation time: 12.2 sec).
<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /><figcaption>Figure 6.4: Histograms of</figcaption></figure>

(a): simulated values at $T = 1$ of the Skew Bessel process of parameters $(\delta, \beta) = (1.2, 0.8)$ with initial value $x_0 = 0$ using Algorithm 1. The quantities (6.1) were approximated using 100-step Riemann approximations.

(b): histogram of simulated values at $T = 1$ of the Reflected Bessel process of parameter $\delta = 1.1$ with initial value $x_0 = 5$ using Algorithm 1 with a uniform grid of step-size $h = 0.01$.

(c): histogram of simulated values at $T = 1$ of the Bessel process of parameter $\delta = 1.8$ with initial value $x_0 = 5$ using Algorithm 1 with a uniform grid of step-size $h = 0.01$. The quantities (6.1) were approximated using 100-step Riemann approximations.
### 6.1.2 Sticky Brownian motion:

The Brownian motion with a sticky point at 0 where the stickiness parameter is $\rho > 0$ can be defined as the diffusion process with scale function and speed measure

$$s(x) = x \quad \text{and} \quad m(dx) = 2dx + \rho\delta_0(dx).$$

From the definition of the scale function,

$$v_0(x) = \frac{x-a}{b-a}.$$

From Proposition 5.3.2 we may deduce the following expressions for the conditional exit times of $c_x = (a, b)$,

$$v_1(x) = v_1^{BM}(x) + \rho \mathbb{1}_{0 \in (a,b)} G_{(a,b)}(x, 0) v_0(0), \\ \bar{v}_1(x) = \bar{v}_1^{BM}(x) + \rho \mathbb{1}_{0 \in (a,b)} G_{(a,b)}(x, 0) (1 - v_0(0)),$$

where $v_1^{BM}(x)$ and $\bar{v}_1^{BM}(x)$ are the analogous quantities for the standard Brownian motion.

### 6.1.3 Skew Brownian motion:

The skew Brownian motion at 0 with parameter $\beta \in (0, 1)$ can be defined as the diffusion process with scale function and speed measure

$$s(x) = \begin{cases} \frac{x}{\beta}, & x \ge 0, \\ \frac{x}{1-\beta}, & x \le 0 \end{cases} \quad \text{and} \quad m(dx) = \begin{cases} 2\beta \, dx, & x > 0, \\ 2(1-\beta) \, dx, & x < 0. \end{cases}$$

From the definition of the scale function,

$$v_0(x) = \frac{s(x) - s(a)}{s(b) - s(a)}.$$

As the $\beta$ and $(1-\beta)$ terms between the speed measure and the scale function compensate themselves in the expressions giving $v_k$ in Proposition 5.3.2,

$$v_1(x) = v_1^{BM}(x) \quad \text{and} \quad \bar{v}_1(x) = \bar{v}_1^{BM}(x).$$

### 6.1.4 Ornstein-Uhlenbeck process:

Let $X_t$ be an Ornstein-Uhlenbeck with mean reversion force $\theta > 0$, long-term mean $\mu$ and diffusion parameter $\sigma > 0$, i.e.

$$dX_t = \theta(\mu - X_t) dt + \sigma dB_t,$$
where $B_t$ is a standard Brownian motion. We will see that $v_0$ can be expressed in terms of the Gaussian imaginary error function as $s(x) = \operatorname{erfi}(\sqrt{\frac{\theta}{\sigma^2}}(\mu - x))$,

$$v_0(x) = \frac{\int_a^x e^{-\int_a^y \frac{2\mu(\zeta)}{\sigma^2(\zeta)} d\zeta} dy}{\int_a^b e^{-\int_a^y \frac{2\mu(\zeta)}{\sigma^2(\zeta)} d\zeta} dy} = \frac{\operatorname{erfi}(\sqrt{\frac{\theta}{\sigma^2}}(\mu - x)) - \operatorname{erfi}(\sqrt{\frac{\theta}{\sigma^2}}(\mu - a))}{\operatorname{erfi}(\sqrt{\frac{\theta}{\sigma^2}}(\mu - b)) - \operatorname{erfi}(\sqrt{\frac{\theta}{\sigma^2}}(\mu - a))},$$

where $\operatorname{erfi}(x) = \frac{2}{\pi} \int_0^x e^{t^2} dt = \frac{2}{\pi} e^{x^2} D_+(x)$, with $D_+(x)$ being Dawson function<fnref n="9" />. Thus, for an Ornstein-Uhlenbeck process (5.44) becomes,

$$v_1(x) = \int_a^b v_0(x \wedge \zeta) (1 - v_0(x \vee \zeta)) v_0(\zeta) c \exp\left(\frac{\theta(\zeta - \mu)^2}{\sigma^2}\right) \frac{2}{\sigma^2} dx,$$

$$\bar{v}_1(x) = \int_a^b v_0(x \wedge \zeta) (1 - v_0(x \vee \zeta)) (1 - v_0(\zeta)) c \exp\left(\frac{\theta(\zeta - \mu)^2}{\sigma^2}\right) \frac{2}{\sigma^2} dx,$$

$$\text{where } c = \operatorname{erfi}\left(\sqrt{\frac{\theta}{\sigma^2}}(\mu - b)\right) - \operatorname{erfi}\left(\sqrt{\frac{\theta}{\sigma^2}}(\mu - a)\right).$$

### 6.1.5 Cox-Ingersoll-Ross process:

The Cox-Ingersoll-Ross process or CIR process [23], introduced first by W. Feller [38], is the diffusion that solves the SDE

$$dX_t = \theta(\mu - X_t) dt + \sigma \sqrt{X_t} dB_t,$$

where $B_t$ is a standard Brownian motion. The parameter $\theta > 0$ expresses its mean reversion speed, $\mu$ its long term speed and $\sigma > 0$ is its diffusivity parameter. The parameter $\theta > 0$ expresses its mean reversion speed, $\mu$ its long term speed and $\sigma > 0$ is its diffusivity parameter. This equation has a diffusion coefficients that degenerates at 0. It however remains non-negative given $X_0 \ge 0$ and almost surely never hit 0 when $2\theta\mu > \sigma^2$. A large body of work have been devoted to the simulation of the CIR and related process, see e.g. [2].

From (5.43) and (5.45),

$$v_0(x) = \frac{\int_a^x y^{-\frac{2\theta\mu}{\sigma^2}} e^{\frac{2\theta}{\sigma^2}y} dy}{\int_a^b y^{-\frac{2\theta\mu}{\sigma^2}} e^{\frac{2\theta}{\sigma^2}y} dy}$$

and

$$v_1(x) = \int_a^b v_0(x \wedge \zeta) (1 - v_0(x \vee \zeta)) \frac{v_0(\zeta)}{v'_0(\zeta) \sigma^2 \zeta} d\zeta,$$

$$\bar{v}_1(x) = \int_a^b v_0(x \wedge \zeta) (1 - v_0(x \vee \zeta)) \frac{1 - v_0(\zeta)}{v'_0(\zeta) \sigma^2 \zeta} d\zeta.$$

As the scale function yield no satisfactory closed formula, we perform a numerical approximation of both $v_0$ and the couple $(v_1, \bar{v}_1)$.

These functions may be computed numerically. One may choose a suitable grid when the process is close to 0 and it is noteworthy that the process may not cross 0, a problem which arise when using Euler type schemes.

<footnote n="9">⁹We use the latter representation in our numerical results.</footnote>
### 6.1.6 Skew and reflected Bessel process:

The Skew-Bessel process (see [3]) of dimension $\delta \in (0, 2)$ and skew $\beta \in (0, 1)$ is the diffusion process with the following scale function and speed measure,

$$s(x) = \begin{cases} \frac{1}{\beta} \frac{x^{2-\delta}}{2-\delta}, & x > 0, \\ -\frac{1}{1-\beta} \frac{|x|^{2-\delta}}{2-\delta}, & x \le 0, \end{cases} \qquad m(dx) = \begin{cases} 2\beta x^{\delta-1} dx, & x > 0, \\ 2(1-\beta)|x|^{\delta-1} dx, & x \le 0. \end{cases}$$

This yields the following expressions for the quantities we need to compute in order to implement the algorithm,

$$\begin{align} v_0(x) &= \frac{s(x) - s(a)}{s(b) - s(a)}, \notag \\ v_1(x) &= \int_a^b G_{a,b}(x, \zeta) v_0(\zeta) 2|\zeta|^{\delta-1} d\zeta, \tag{6.1} \\ \bar{v}_1(x) &= \int_a^b G_{a,b}(x, \zeta) (1 - v_0(\zeta)) 2|\zeta|^{\delta-1} d\zeta, \notag \end{align}$$

where $G_{a,b}(x, \zeta)$ is defined in (1.20). The probability transition kernels plotted in Figure 6.3 are computed in [3].

## 6.2 Local time approximation

In this section we present the results of numerical simulations regarding the local time approximation developed in Chapter 4. With numerical experiments, we assess the asymptotic properties of the local time approximation (4.7). Also, with an example we illustrate the flexibility of Space-Time Markov Chain Approximations generated via Algorithm 1. One feature of such approximation processes is that they are defined on a given grid. With a suitable choice of grid it is possible to achieve higher orders of convergence of localized path-sensitive functionals like (4.7).

We simulate trajectories of an approximation process of the sticky Brownian motion of parameter $\rho$. Then, we compare the stickiness parameter estimations (4.8) with the true value of $\rho$. For the numerical simulations we use the Space-time Markov chain approximation or STMCA Algorithm [6]. This algorithm uses grid-valued random walks to approximate the law of any one-dimensional generalized diffusion process. The STMCA Algorithm is particularly adapted to our problematic as:

* It is well suited for the simulation of sticky singular one-dimensional diffusions.

* By suitable choice of the grid, we can control the amount of path-observations of $X$ observed through the test function $g$ in (4.7). In particular, for grids that satisfy the condition in Corollary 2.5 of [6], the convergence speed of the STMCA algorithm is optimal. Thus, if one increases the precision of such a grid $g$ around 0, we approximate accurately $X$ with $g$-valued STMCA random walks and feed more path-observations to the statistic (4.7) without paying a too heavy computational cost.
**The statistic:** Let $X$ be a sticky Brownian motion of parameter $\rho$ and $g$ the function defined for every $x \in \mathbb{R}$ by $g(x) = \mathbb{1}_{1<|x|<5}/8$. For every $\alpha \in (0, 1)$ and $n \in \mathbb{N}$, we define the following path-wise statistics of $X$:

$$ T_{n,\alpha}^{(1)}(X) := \frac{n^{\alpha}}{n} \sum_{i=1}^{[nt]} g(n^{\alpha} X_{\frac{i-1}{n}}^{\rho}), \qquad (6.2) $$

$$ T_{n,\alpha}^{(2)}(X) := \frac{2}{n} \frac{\sum_{i=1}^{[nt]} \mathbb{1}_{X_{\frac{i-1}{n}}=0}}{T_{n,\alpha}^{(1)}(X)}. \qquad (6.3) $$

From (4.7), for any $\alpha \in (0, 1/2)$, the statistics (6.2) and (6.3) converge to $L_t^0(X)$ and $\rho$ respectively in probability. We use (6.3) as proxy to assess the properties of the local time approximation.

**The grids:** For every $h > 0$, let $\mathbf{g}_0$ and $\mathbf{g}_1$ be two grids defined by:

$$ \mathbf{g}_0(h) = \{0\} \cup \{\pm (h^2/\rho + kh); k \in \mathbb{N}\}, \quad \mathbf{g}_1(h) = \{\pm x_k(h); k \ge 0\}, $$

where $\{x_j(h)\}_{j \ge 0}$ is defined recursively by

$$ x_0 = 0, \quad x_j = x_{j-1} + \left( \frac{h^2}{\rho} \frac{1}{x_{j-1} + 1} + h \left( 1 - \frac{1}{x_{j-1} + 1} \right) \right) \mathbb{1}_{\{x_{j-1} < 1\}} + h \mathbb{1}_{\{x_{j-1} \ge 1\}}. $$

We observe that

$$ |\mathbf{g}_1| = |\mathbf{g}_0| \quad \text{and} \quad |\mathbf{g}_1|_X = |\mathbf{g}_0|_X. $$

From Theorem 5.2.1, the asymptotic convergence of the STMCA is $O(h^{1/2})$ for both grids.

<table><caption>Table 6.1: Stickiness parameter estimations using the grid $\mathbf{g}_0$ for $h = 0.01$. The missing values in the table corresponds to cases where the statistic (6.3) is observed to be 0. Computation time (single-core): 4 seconds.</caption><thead><tr><th>α</th><th>n</th><th>ρ̂<sub>MC</sub></th><th>Ŝ<sup>2</sup><sub>MC</sub></th><th>σ̂<sub>MC</sub></th><th>acc̄</th><th>rej/N<sub>MC</sub></th></tr></thead><tbody><tr><td>0.3</td><td>100000</td><td>1.33451</td><td>0.740592</td><td>0.860577</td><td>0.125709</td><td>0/2000</td></tr><tr><td>0.4</td><td>100000</td><td>1.06132</td><td>0.142955</td><td>0.378094</td><td>0.0460073</td><td>0/2000</td></tr><tr><td>0.5</td><td>100000</td><td>1.3148</td><td>0.162762</td><td>0.403438</td><td>0.0116839</td><td>0/2000</td></tr><tr><td>0.55</td><td>100000</td><td></td><td></td><td></td><td></td><td>2000/2000</td></tr><tr><td>0.6</td><td>100000</td><td></td><td></td><td></td><td></td><td>2000/2000</td></tr><tr><td>0.65</td><td>100000</td><td></td><td></td><td></td><td></td><td>2000/2000</td></tr></tbody></table>

Within each of the following tables we use the same simulated STMCA trajectories of the sticky Brownian motion of parameter $\rho = 1$:

**Simulation metrics:** The integer $N_{\text{MC}}$ will be the Monte Carlo simulation size. For every $j \le N_{\text{MC}}$ we simulate the path of an approximation process $X_t^j$. To assess the quality of each Monte Carlo estimation we use the following metrics:
<figure><img src="image_25.png" /><figcaption>Figure 6.5: Stickiness parameter estimations histograms using (6.3) with $n = 100000$ and $\alpha = 0.3$. (true value $\rho = 1$)</figcaption></figure>

• $(\hat{\rho}_{MC}, \hat{S}_{MC}^2, \hat{\sigma}_{MC})$: Monte Carlo estimation, variance and standard deviation of the estimated stickiness parameters,

• $\widehat{acc}$: average number of path-values observed by $g$, i.e

$$ \widehat{acc} = \frac{1}{N_{MC}} \sum_{j=1}^{N_{MC}} \frac{1}{n} \sum_{i=1}^{n} \mathbb{1}_{g(X_{(i-1)/n}^{j}) \neq 0} $$

• rej: percentage of trajectories where the local time estimation equals 0, i.e

$$ \text{rej} = \# \left\{ j \le N_{MC} : \frac{n^{\alpha}}{n} \sum_{i=1}^{[nt]} g(n^{\alpha} X_{\frac{i-1}{n}}^{j}) = 0 \right\}. $$

### 6.2.1 Observations, conclusion

From the numerical experiments (Tables 6.1-6.5), we observe the following:

• As long as $n$ is high enough, the higher the value of $\alpha \in (0, 1)$, the lower the estimated variance and the better the approximation of the local time,

• Also, the higher the $\alpha$, the more the trajectory of $X^\rho$ is inflated and the less things are observed through $g$. Thus, having a finite set of path-wise observations of $X^\rho$, one must find an $\alpha \in (0, 1)$ large enough to trigger the asymptotic regime of (4.1) and low enough so we do not dump too much path-wise observations.
<figure><img src="image_26.png" /><figcaption>Figure 6.6: Stickiness parameter estimations histograms using (6.3) with $n = 100000$ and $\alpha = 0.55$. (true value $\rho = 1$)</figcaption></figure>

* In Table 6.5 we see that for a fixed $c > 0$, every $(n, \alpha)$ such that $\log n = c/\alpha$ yield the same Monte Carlo variance. This relation can be guessed from (4.4) and (4.1),

* The convergence (4.7) seems to hold for $\alpha \in [1/2, 1)$, values not covered by Theorem 4.1.3. We thus conjecture the following:

Algorithm 1 gives us the flexibility to remediate to the latter by using grids of higher precision around the point of stickiness and thus achieving higher orders of convergence for (6.2) without significant increase in the numerical complexity.

We observe that the usage of grid $g_1$ yields far superior results than $g_0$. Using $g_1$ we have an abundance of simulated path-wise observations close to the point of stickiness. The statistic (6.2) remains thus relevant for large values of $\alpha$ and we can achieve higher orders of convergence.

**Conjecture 6.2.1.** Theorem 4.1.3 holds for any $\alpha \in (0, 1)$ and the rate of convergence of both (4.8) and (4.7) are increasing in terms of $\alpha$.
<table><caption>Table 6.2: Stickiness parameter estimations using the grid $g_1$ for $h = 0.01$. Computation time (single-core): 67 seconds.</caption><thead><tr><th>α</th><th>n</th><th>ρ̂<sub>MC</sub></th><th>Ŝ<sup>2</sup><sub>MC</sub></th><th>σ̂<sub>MC</sub></th><th>acĉ</th><th>rej/N<sub>MC</sub></th></tr></thead><tbody><tr><td>0.3</td><td>100000</td><td>1.25282</td><td>0.629184</td><td>0.793211</td><td>0.127209</td><td>0/2000</td></tr><tr><td>0.4</td><td>100000</td><td>1.06863</td><td>0.18705</td><td>0.432493</td><td>0.0437251</td><td>0/2000</td></tr><tr><td>0.5</td><td>100000</td><td>1.02367</td><td>0.0721315</td><td>0.268573</td><td>0.0140703</td><td>0/2000</td></tr><tr><td>0.55</td><td>100000</td><td>1.01813</td><td>0.0672543</td><td>0.259334</td><td>0.00796982</td><td>0/2000</td></tr><tr><td>0.6</td><td>100000</td><td>1.01266</td><td>0.031502</td><td>0.177488</td><td>0.00448427</td><td>0/2000</td></tr><tr><td>0.65</td><td>100000</td><td>1.01594</td><td>0.0268068</td><td>0.163728</td><td>0.00251455</td><td>2/2000</td></tr></tbody></table>

<table><caption>Table 6.3: Simulation results for fixed $α$ and different values of $n$.</caption><thead><tr><th>α</th><th>n</th><th>ρ̂<sub>MC</sub></th><th>Ŝ<sup>2</sup><sub>MC</sub></th><th>σ̂<sub>MC</sub></th><th>acĉ</th><th>rej/N<sub>MC</sub></th></tr></thead><tbody><tr><td>0.6</td><td>20000</td><td>1.04379</td><td>0.14514</td><td>0.380973</td><td>0.0121671</td><td>0/1000</td></tr><tr><td>0.6</td><td>40000</td><td>1.02967</td><td>0.03178</td><td>0.178288</td><td>0.0079693</td><td>0/1000</td></tr><tr><td>0.6</td><td>100000</td><td>1.00135</td><td>0.01592</td><td>0.126199</td><td>0.0046799</td><td>0/1000</td></tr><tr><td>0.6</td><td>300000</td><td>1.00638</td><td>0.00850</td><td>0.092235</td><td>0.0023960</td><td>0/1000</td></tr></tbody></table>

<table><caption>Table 6.4: Simulation results for fixed $n$ and different values of $α$. Estimation with an asterisk were performed removing the trajectories with $T_n^{(1)}(X) = 0$.</caption><thead><tr><th>α</th><th>n</th><th>ρ̂<sub>MC</sub></th><th>Ŝ<sup>2</sup><sub>MC</sub></th><th>σ̂<sub>MC</sub></th><th>acĉ</th><th>rej/N<sub>MC</sub></th></tr></thead><tbody><tr><td>0.3</td><td>100000</td><td>1.28303</td><td>0.66712</td><td>0.81677</td><td>0.126344</td><td>0/2000</td></tr><tr><td>0.4</td><td>100000</td><td>1.12398</td><td>0.38812</td><td>0.62299</td><td>0.043843</td><td>0/2000</td></tr><tr><td>0.5</td><td>100000</td><td>1.04014</td><td>0.09153</td><td>0.30253</td><td>0.014309</td><td>0/2000</td></tr><tr><td>0.6</td><td>100000</td><td>1.01083*</td><td>0.04976*</td><td>0.22307*</td><td>0.004598*</td><td>1/2000</td></tr><tr><td>0.7</td><td>100000</td><td>1.06961*</td><td>0.14296*</td><td>0.37810*</td><td>0.001367*</td><td>13/2000</td></tr><tr><td>0.8</td><td>100000</td><td>1.01483*</td><td>0.12386*</td><td>0.35194*</td><td>0.000467*</td><td>23/2000</td></tr></tbody></table>

<table><caption>Table 6.5: Simulation results for $(n, α)$ satisfying $\log n = 5.5/α$.</caption><thead><tr><th>α</th><th>n</th><th>ρ̂<sub>MC</sub></th><th>Ŝ<sup>2</sup><sub>MC</sub></th><th>σ̂<sub>MC</sub></th><th>acĉ</th><th>rej/N<sub>MC</sub></th></tr></thead><tbody><tr><td>0.55536</td><td>20000</td><td>1.04813</td><td>0.094319</td><td>0.307115</td><td>0.0190297</td><td>0/1000</td></tr><tr><td>0.519033</td><td>40000</td><td>1.04893</td><td>0.095243</td><td>0.308616</td><td>0.0190053</td><td>0/1000</td></tr><tr><td>0.477724</td><td>100000</td><td>1.04946</td><td>0.100884</td><td>0.317623</td><td>0.0190204</td><td>0/1000</td></tr><tr><td>0.436109</td><td>300000</td><td>1.04931</td><td>0.096736</td><td>0.311024</td><td>0.0190139</td><td>0/1000</td></tr></tbody></table>
# Appendix A

## A.1 Strong continuity of diffusion semi-groups on $C_b(\mathbb{I})$: a counter-example

**Proposition A.1.1.** Let $(P_t)_{t \ge 0}$ be the semi-group of the standard Brownian motion $B$, defined for every measurable bounded $f$, $x \in \mathbb{R}$ and $t \ge 0$ by

$$P_t f(x) = E_x(f(B_t)).$$

Then, $(P_t)_{t \ge 0}$ is not strongly continuous on $C_b(\mathbb{R})$.

*Proof.* Let $\phi_n$ be the sequence of functions defined for every $n \in \mathbb{N}$ and $x \in \mathbb{R}$ by

$$\phi_n(x) = (1 - n|x|)1_{|x|<1/n}.$$

Let also $f$ and $g$ be the functions defined for every $x \in \mathbb{R}$ by

$$f(x) = \sum_{n \in \mathbb{N}} \phi_n(x - n^2)$$

and

$$g(x) = 1_{|x-n^2|<1/n} + 1_{|x-n^2|>\Delta_n},$$

where $\Delta_n = n^2 - (n-1)^2 - 1/n = 2n - 1 - 1/n$. We observe that for all $n \in \mathbb{N}$,

$$f(n^2) = 1. \qquad (A.1)$$

Also, if $Z \sim \mathcal{N}(0, 1)$,

$$P_t f(n^2) = E_{n^2} \left( f(\sqrt{t}Z) \right) \le E \left( g(n^2 + \sqrt{t}Z) \right) \\ = P \left( Z \in \left( -\frac{1}{n\sqrt{t}}, \frac{1}{n\sqrt{t}} \right) \right) + P \left( |Z| \ge \Delta_n/\sqrt{t} \right).$$

Thus, for all $t \ge 0$ and $\epsilon > 0$, there exists $n_0 \in \mathbb{N}$ large enough such that for all $n \ge n_0$,

$$P_t f(n^2) < \epsilon. \qquad (A.2)$$

From (A.1) and (A.2), for all $t \ge 0$,

$$\sup_{x \in \mathbb{R}} |P_t f(x) - f(x)| \ge 1$$

and thus $(P_t)_{t \ge 0}$ is not strongly continuous over $C_b(\mathbb{R})$.

□
# A.2 Semimartingale results

## A.2.1 Change of variables in a stochastic integral

**Definition A.2.1.** • An $(\mathcal{F}_t)_{t \ge 0}$-time change is any almost surely increasing, right-continuous family of $(\mathcal{F}_t)_{t \ge 0}$-stopping times $(T_t)_{t \ge 0}$ such that $T_0 = 0$.

• A process $M$ is said to be in synchronization with $T$ iff $M$ is constant on $[T_{s-}, T_s]$.

• Given an $(\mathcal{F}_t)_{t \ge 0}$-semi-martingale $M$, let $L(M, (\mathcal{F}_t)_{t \ge 0})$ be the class of $(\mathcal{F}_t)_{t \ge 0}$-progressively measurable, predictable càdlag processes $H$ for which the integral $\int_0^t H_s dM_s$ can be constructed.

**Theorem A.2.2** (Theorem 3.1 of [64], Proposition 10.21 of [54]). Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ be a filtered probability space, $M$ an $(\mathcal{F}_t)_{t \ge 0}$-semimartingale in synchronization with an $(\mathcal{F}_t)_{t \ge 0}$-time change $T$. If $H \in L(Z, (\mathcal{F}_t)_{t \ge 0})$, then $(H_{T_{t-}})_{t \ge 0} \in L(Z \circ T, (\mathcal{F}_{T_t})_{t \ge 0})$. Also,

$$ \int_0^{T_t} H_s \, dM_s = \int_0^t H_{T_{s-}} \, dM_{T_s}, \qquad (A.3) $$

for every $t \ge 0$.

## A.2.2 Local time scaling

**Lemma A.2.3** (Exercise 1.23 of [80]-Chapter VI). Let $X$ be a continuous semi-martingale and $f$ a strictly increasing difference of two convex functions. If $f(X) = (f(X_t))_{t \ge 0}$, then for every $a \in \mathbb{I}$ and $t \ge 0$,

$$ L_t^{f(a)}(f(X)) = f'(a)L_t^a(X) $$

almost surely, where $f'$ is the right-derivative of $f$.

**Lemma A.2.4** (Exercise 1.27 of [80]-Chapter VI). Let $X$ be a semi-martingale with state-space $\mathbb{I}$, $T$ a time change and $Y$ is the time changed process such that $Y_t = X_{T(t)}$ for every $t \ge 0$. Then for every $a \in \mathbb{I}$ and $t \ge 0$,

$$ L_t^a(Y) = L_{T(t)}^a(X) \qquad (A.4) $$

almost surely.

## A.2.3 Stochastic Dini's theorem

**Definition A.2.5.** A sequence of processes $X^n$ is said to converge locally uniformly in time, in P-probability to a process $X$ iff

$$ \sup_{s \le t} \{|X_s^n - X_s|\} \xrightarrow{n \to \infty} 0, $$

in P-probability, for every $t \ge 0$
**Lemma A.2.6** ((2.2.16) of [56]). Let {$X^n$}$_{n \in \mathbb{N}}$ be a sequence of increasing processes and X an almost surely continuous bounded process. If {$X^n$}$_{n \in \mathbb{N}}$ and X are defined on the probability space $(\Omega, \mathcal{F})_{t \ge 0}$, P) and

$$X_t^n \xrightarrow{n \to \infty} X_t$$

in probability, for every $t \in D$ with D a dense subset of $\mathbb{R}_+$. Then, the convergence is locally uniform in time, in probability.

## A.3 Joint scaling of the sticky Brownian motion

**Lemma A.3.1.** Let {($\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}$, $P_x^\rho$); $x \in \mathbb{R}, \rho \ge 0$} be a family of filtered probability spaces and $X^\rho = (X_t^\rho)_{t \ge 0}$ a process defined on $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0})$ such that under $P_x^\rho$ it is the sticky Brownian motion of stickiness parameter $\rho$ and $P_x^\rho(X_0^\rho = x) = 1$. Then,

$$ \begin{aligned} & \mathrm{Law}_{P_x^\rho} (X_{ct}^\rho, L_{ct}^0(X^\rho), A_{ct}^+(X^\rho); t \ge 0) \\ & = \mathrm{Law}_{P_x^{\rho/\sqrt{c}}} (\sqrt{c}X_t^{\rho/\sqrt{c}}, \sqrt{c}L_t^0(X^{\rho/\sqrt{c}}), cA_t^+(X^{\rho/\sqrt{c}}); t \ge 0), \quad (\text{A.5}) \end{aligned} $$

where $P_x^\rho(X_0^{\rho/\sqrt{c}} = \sqrt{c}x) = 1$ and $(L^0(X^\rho), A^+(X^\rho))$ and $(L^0(X^{\rho/\sqrt{c}}), A^+(X^{\rho/\sqrt{c}}))$ are the local times, occupation times<fnref n="10" /> of $\mathbb{R}_+$ pairs of $X^\rho$ and $X^{\rho/\sqrt{c}}$ respectively.

*Proof.* From [88], the joint density of $(X_t^\rho, L_t^0(X^\rho), A_{ct}^+(X^\rho))$ is defined for every $(t, x, y, l, o) \in \mathbb{R}_+ \times \mathbb{R}^2 \times [0, \frac{2}{\rho}] \times [0, t]$ by

$$ \begin{aligned} & P_x(X_t^\rho \in dy, L_t^0(X^\rho) \in d\ell, A_{ct}^+(X^\rho) \in do) \\ & = q_\rho(t, x, y, l, o) dy dl do = h\left(o - \rho\ell, \frac{\ell}{2} + x\right) h\left(t - o, \frac{\ell}{2} - y\right) dy dl do, \end{aligned} $$

where $h(t, x)$ is the function defined for every $t \ge 0$ and $x \in \mathbb{R}$ by

$$h(t, x) = \frac{|x|}{\sqrt{2\pi t^{3/2}}} e^{-x^2/2t}.$$

We observe that

$$h(ct, x) = c^{-1}h(t, x/\sqrt{c}).$$

Thus,

$$ \begin{aligned} q_\rho(ct, x, y, l, o) &= h\left(o - \rho\ell, \frac{\ell}{2} + x\right) h\left(ct - o, \frac{\ell}{2} - y\right) \\ &= h\left(c\left(\frac{o}{c} - \frac{\rho\ell}{c}\right), \frac{\ell}{2} + x\right) h\left(c\left(t - \frac{o}{c}\right), \frac{\ell}{2} - y\right) \\ &= c^{-2}h\left(\frac{o}{c} - \frac{\rho\ell}{c}, \frac{\ell}{2\sqrt{c}} + \frac{x}{\sqrt{c}}\right) h\left(t - \frac{o}{c}, \frac{\ell}{2\sqrt{c}} - \frac{y}{\sqrt{c}}\right) \\ &= c^{-2}q_{\rho/\sqrt{c}}\left(t, \frac{x}{\sqrt{c}}, \frac{y}{\sqrt{c}}, \frac{l}{\sqrt{c}}, \frac{o}{c}\right) \end{aligned} $$

$$A_t^B(X) = \int_0^t 1_{X_s \in B} \mathrm{d}s.$$

<footnote n="10">¹⁰The occupation time by X of a measurable set $B \in \mathcal{B}(\mathbb{R})$ is the process $A^B(X)$ defined for every $t \ge 0$ by</footnote>
and

$$q_{\rho}(ct, x, y, l, o) \, dy \, d\ell \, do = q_{\rho/\sqrt{c}}\left(t, \frac{x}{\sqrt{c}}, \frac{y}{\sqrt{c}}, \frac{l}{\sqrt{c}}, \frac{o}{c}\right) d\left(\frac{y}{\sqrt{c}}\right) d\left(\frac{l}{\sqrt{c}}\right) d\left(\frac{o}{c}\right),$$

which finishes the proof. □

## A.4 Conditioning on the embedded path of a diffusion

Let $X$ be a diffusion process with state-space $\mathbb{I}$ an interval of $\mathbb{R}$ defined on a family of probability spaces $\mathcal{P} = (\Omega, (\mathcal{F}_t)_{t \ge 0}, P_x)_{x \in \mathbb{I}}$ such that for every $x \in \mathbb{I}$, $P_x(X_0 = x) = 1$. Let also $\mathbf{g}$ be a covering grid of $\mathbb{I}$. We define the embedding times of $(X_t)_{t \ge 0}$ in $\mathbf{g}$ as,

$$\left\{ \begin{array}{l} \tau_0^{\mathbf{g}} = 0, \\ \tau_k^{\mathbf{g}} = \inf \{ t > \tau_{k-1}^{\mathbf{g}} : X_t \in \mathbf{g} \setminus \{ X_{\tau_{k-1}}^{\mathbf{g}} \} \}, \quad \forall k \ge 1. \end{array} \right. \qquad (A.6)$$

**Lemma A.4.1.** Let $X$ be a diffusion process with state-space $\mathbb{I}$ an interval of $\mathbb{R}$ defined on a family of probability spaces $\mathcal{P} = (\Omega, (\mathcal{F}_t)_{t \ge 0}, P_x)_{x \in \mathbb{I}}$ such that for every $x \in \mathbb{I}$, $P_x(X_0 = x) = 1$. Let also $\mathbf{g}$ be a covering grid of $\mathbb{I}$, $\{\tau_j\}_{j \in \mathbb{Z}_+}$ the sequence of embedding times of $X$ in $\mathbf{g}$ defined in (A.6) and $\mathcal{B}$ the sigma-algebra defined by $\mathcal{B} = \sigma\{X_{\tau_j}; j \in \mathbb{N}_0\}$. Then, for any measurable path-functional $F: C^0(\mathbb{R}_+, \mathbb{I}) \to \mathbb{R}$ and $j \ge 1$ and $x \in \mathbb{I}$,

$$E_x(F((X_t^{\tau_{j-1}, \tau_j})_{t \ge 0}) | \mathcal{B}) = E_x(F((X_t^{\tau_{j-1}, \tau_j})_{t \ge 0}) | X_{\tau_{j-1}}, X_{\tau_j}),$$

where $X^{\tau_{j-1}, \tau_j}$ is the process defined for every $t \ge 0$ by $X_t^{\tau_{j-1}, \tau_j} = X_{(\tau_{j-1}+t) \wedge \tau_j}$.

*Proof.* Let us fix $x_1, x_2, \dots$ be a sequence of points in the grid. Let us define

$$Q(x; x_1, x_2, \dots) := P_x \left( X_{\tau_1} = x_1, X_{\tau_2} = x_2, \dots \right)$$

By the strong Markov property,

$$\begin{align*} P_x \left( X_{\tau_{i+1}} = x_{i+1}, X_{\tau_{i+2}} = x_{i+2}, \dots \middle| \mathcal{F}_{\tau_i} \right) &= P_{X_{\tau_i}} \left( X_{\tau_1} = x_{i+1}, \dots, X_{\tau_{i+2}} = x_{i+2} \right) \\ &= Q(X_{\tau_i}; x_{i+1}, x_{i+2}, \dots). \end{align*}$$

Using the strong Markov property twice, first by conditioning first with respect to $\mathcal{F}_{\tau_{j+1}}$ and then with respect to $\mathcal{F}_{\tau_j}$,

$$\begin{align*} E_x \left( \mathbb{1}_{X_{\tau_1} = x_1, \dots, X_{\tau_j} = x_j} F\left(\left(X_t^{\tau_{j-1}, \tau_j}\right)_{t \ge 0}\right) \mathbb{1}_{X_{\tau_j} = x_j, X_{\tau_{j+1}} = x_{j+1}, \dots} \right) \\ &= E_x \left( \mathbb{1}_{X_{\tau_1} = x_1, \dots, X_{\tau_j} = x_j} F\left(\left(X_t^{\tau_{j-1}, \tau_j}\right)_{t \ge 0}\right) \mathbb{1}_{X_{\tau_j} = x_j} Q(X_{\tau_j}) \right) \\ &= E_x \left( \mathbb{1}_{X_{\tau_1} = x_1, \dots, X_{\tau_{j-1}} = x_{j-1}} \left(X_t^{\tau_{j-1}, \tau_j}\right)_{t \ge 0} \right) Q(x_j) = E_x \left( \mathbb{1}_{X_{\tau_1} = x_1, \dots, X_{\tau_j} = x_{j-1}} \right) R(x_{j-1}, x_j) Q(x_j) \end{align*}$$
with $R(x, y) := P_x(F((X_t^{0, \tau_1})_{t \ge 0})1_{X_{\tau_1} = y})$. Therefore, using the definition of the conditional expectation,

$$ \begin{aligned} & E_x \left( F((X_t^{\tau_{j-1}, \tau_j})_{t \ge 0}) \middle| X_{\tau_1} = x_1, X_{\tau_2} = x_2, \dots \right) \\ & = \frac{R(x_{j-1}, x_j)}{P_{x_{j-1}}(X_{\tau_1} = x_j)} = E_{x_{j-1}} \left( F((X_t^{0, \tau_1})_{t \ge 0}) \middle| X_{\tau_1} = x_j \right). \end{aligned} $$

This is sufficient to prove the result. $\square$

<figure><img src="image_27.png" /></figure>
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