Importance sampling in path space for diffusion processes with slow-fast variables

Abstract

Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton–Jacobi–Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.

Appendices

Two useful inequalities

Claim 7.1

Consider functions $x_1(t), x_2(t)$ on $t \in [0,T]$ satisfying

$$\begin{aligned} {\dot{x}}_1(t)&\le a_{11}\,x_1(t) + a_{12}\,x_2(t) \\ {\dot{x}}_2(t)&\le \frac{a_{21}}{\epsilon } x_1(t) - \frac{a_{22}}{\epsilon } x_2(t) \end{aligned}$$

with $x_1(0) = 0, x_2(0) = 1$, $a_{ij} > 0$, $1\le i, j \le 2$. Further assume that $x_1(t) \ge 0$ for all $t\in [0,T]$. Then there is a constant $C > 0$ depending on $a_{ij}$ and T, such that

$$\begin{aligned} \max _{0\le s \le T} x_1(s) \le C\epsilon \,, \qquad x_2(t) \le e^{-\frac{a_{22} t}{\epsilon }} + C\epsilon \,, \quad t \in [0, T]. \end{aligned}$$

(7.1)

Proof

Applying Gronwall’s inequality to the equation of $x_2$, we have

$$\begin{aligned} x_2(t)&\le e^{-\frac{a_{22}t}{\epsilon }} + \int _0^t e^{-\frac{a_{22}}{\epsilon }(t-s)} \frac{a_{21}}{\epsilon } x_1(s) ds \nonumber \\&\le e^{-\frac{a_{22}t}{\epsilon }} + \frac{a_{21}}{a_{22}} \max _{0 \le s \le t} x_1(s)\,. \end{aligned}$$

(7.2)

Applying Gronwall’s inequality to $x_1$ and using (7.2), we find

$$\begin{aligned} x_1(t) \le a_{12} \int _0^t e^{a_{11}(t-s)} \Big [e^{-\frac{a_{22}s}{\epsilon }} + \frac{a_{21}}{a_{22}} \big (\max _{0 \le r \le s} x_1(r)\big )\Big ] ds\,. \end{aligned}$$

(7.3)

Since the right hand side in the last inequality is monotonically increasing (as a function of t), it follows that

$$\begin{aligned} \max _{0\le s \le t} x_1(s)&\le a_{12} \int _0^t e^{a_{11}(t-s)} \Big [e^{-\frac{a_{22}s}{\epsilon }} + \frac{a_{21}}{a_{22}} \big (\max _{0 \le r \le s} x_1(r)\big )\Big ] ds \nonumber \\&\le \frac{a_{12}}{a_{22}} e^{a_{11}T}\epsilon + \frac{a_{12}a_{21}}{a_{22}} \int _0^t e^{a_{11}(t-s)} \big (\max _{0\le r \le s} x_1(r)\big ) ds\,. \end{aligned}$$

(7.4)

The first part of the assertion then follows by applying Gronwall’s inequality in integral form to $\max \limits _{0\le s \le t} x_1(s)$, while the second part is obtained using (7.2). $\square$

For $0< \epsilon < 1$, we set $t_1 = -\frac{2\epsilon \ln \epsilon }{\lambda } > 0$ and introduce the function $\gamma :[0,T] \rightarrow [0, 1]$ by

$$\begin{aligned} \gamma (t) =\left\{ \begin{array}{cl} 1- \frac{t}{t_1} &{} \qquad 0 \le t \le t_1 \\ 0 &{} \qquad t_1 < t \le T\,. \end{array} \right. \end{aligned}$$

(7.5)

Claim 7.2

Consider functions $x_1(t), x_2(t)$ on $t \in [0, T]$ satisfying

$$\begin{aligned} {\dot{x}}_1(t)&\le a_1(1 + \epsilon ^{-\gamma (t)}) x_1(t) + a_2\epsilon ^{\gamma (t)} x_2(t) \\ {\dot{x}}_2(t)&\le \frac{a_3x_1(t)}{\epsilon } -\frac{\lambda x_2(t)}{\epsilon } \,, \end{aligned}$$

where $\gamma$ is given in (7.5), $a_i \ge 0, 1\le i \le 3$, and $x_1(0) = 0, x_2(0) = 1$. Further assume that $x_1(t) \ge 0$ on $t \in [0,T]$. Then there is a constant $C > 0$ independent of $\epsilon$, such that

$$\begin{aligned} \max _{0\le s \le T} x_1(s) \le C\epsilon ^2, \qquad x_2(t) \le e^{-\frac{\lambda t}{\epsilon }} + C\epsilon ^2\,, \quad t \in [0,T]\,. \end{aligned}$$

(7.6)

Proof

As in Claim 7.1, we can obtain

$$\begin{aligned} x_2(t)&\le e^{-\frac{\lambda t}{\epsilon }} + \frac{a_3}{\lambda } \max _{0 \le s \le t} x_1(s) \end{aligned}$$

(7.7)

$$\begin{aligned} \max _{0\le s \le t} x_1(s)&\le a_2\int _0^t e^{a_1\int _s^t (1 + \epsilon ^{-\gamma (r)}) dr} \epsilon ^{\gamma (s)} \big [e^{-\frac{\lambda s}{\epsilon }} + \frac{a_3}{\lambda }\big (\max _{0 \le r \le s} x_1(r)\big )\big ] ds \,. \end{aligned}$$

(7.8)

Then, for $t < t_1$, the second inequality above implies

$$\begin{aligned} \max _{0\le s \le t} x_1(s) \le C \epsilon ^2 + \frac{a_2a_3}{\lambda } \int _0^{t} e^{a_1\int _s^t (1 + \epsilon ^{-\gamma (r)}) dr} \epsilon ^{\gamma (s)} \big (\max _{0 \le r \le s} x_1(r)\big ) ds \,. \end{aligned}$$

(7.9)

Using (7.7) and Gronwall’s inequality again, we conclude that

$$\begin{aligned} \max _{0\le s \le t_1} x_1(s) \le C\epsilon ^2, \quad x_2(t) \le e^{-\frac{\lambda t}{\epsilon }} + C\epsilon ^2 , \qquad t \le t_1\,. \end{aligned}$$

(7.10)

Repeating the above argument for $t\in [t_1, T]$, noticing that $x_1(t_1) \le C\epsilon ^2$, $x_2(t_1) \le C\epsilon ^2$, $\gamma (t) \equiv 0, t \in [t_1, T]$, it follows that

$$\begin{aligned} \max _{t_1\le s \le T} x_1(s) \le C\epsilon ^2, \quad x_2(t) \le C\epsilon ^2 , \qquad t \in [t_1, T]\,. \end{aligned}$$

(7.11)

The proof is completed by combining (7.10) and (7.11). $\square$

Properties of the stationary process

For fixed $x \in {\mathbb {R}}^k$ and $\tau \in {\mathbb {R}}$, we introduce the process

$$\begin{aligned} d\xi ^x_{\tau , s} = \frac{1}{\epsilon } g(x,\xi ^x_{\tau , s}) ds + \frac{1}{\sqrt{\epsilon }}\alpha _2(x,\xi ^x_{\tau , s}) dw_s\,, \quad s\ge \tau \,, \quad \xi ^x_{\tau , \tau } = y \end{aligned}$$

(8.1)

where $w_s$ is a standard Wiener process in ${\mathbb {R}}^{m_2}$. In the following, we summarize some properties related to the above process that we called the fast subsystem in Sect. 3. See also [10, 33] for additional results.

Lemma 8.1

Under Assumptions 1–2, there exists a constant $C > 0$, independent of $\epsilon , x, y$, such that:

• 1. ${{\mathbf {E}}}|\xi ^x_{\tau , s}|^4 \le e^{-\frac{\lambda (s-\tau )}{\epsilon }} |y|^4 + C\big (|x|^4 + 1\big )$.

• 2. For $\tau _1 \le \tau _2$, it holds

  $$\begin{aligned}{{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4 \le C\big (|x|^4 + |y|^4+1\big )\,e^{-\frac{4\lambda (s-\tau _2)}{\epsilon }}\,, \quad s \ge \tau _2\,. \end{aligned}$$

• 3. For $x, x' \in {\mathbb {R}}^k$ and $\tau _1 \le \tau _2$,

  $$\begin{aligned} {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4\le e^{-\frac{2\lambda (s-\tau _2)}{\epsilon }} \big (|x|^4 + |y|^4 + 1\big ) + C |x' - x|^4\,, \quad s \ge \tau _2\,. \end{aligned}$$

Proof

• 1. By Ito’s formula, we have

  $$\begin{aligned} \frac{d{{\mathbf {E}}}|\xi ^x_{\tau , s}|^4}{ds} =&\, \frac{1}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s}|^2 \big (4\langle g(x, \xi ^x_{\tau , s}), \xi ^x_{\tau , s}\rangle + 2\Vert \alpha _2(x,\xi ^x_{\tau , s})\Vert ^2\big ) \\&+\, 4|\alpha ^T_2(x,\xi ^x_{\tau , s})\xi ^x_{\tau , s}|^2\Big ] \nonumber \\ \le&\, \frac{1}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s}|^2 \big (4\langle g(x, \xi ^x_{\tau , s}), \xi ^x_{\tau , s}\rangle + 6\Vert \alpha _2(x,\xi ^x_{\tau , s})\Vert ^2\big )\Big ]\,. \end{aligned}$$

  Applying inequality (3.13) in Remark 2 and inequality (5.9), we obtain

  $$\begin{aligned} \frac{d{{\mathbf {E}}}|\xi ^x_{\tau , s}|^4}{ds} \le&-\frac{2\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau , s}|^4 + \frac{C}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s}|^2 (|x|^2 + 1)\Big ] \nonumber \\ \le&-\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau , s}|^4 + \frac{C}{\epsilon } \Big (|x|^4 + 1\Big )\,, \end{aligned}$$

  and the first statement follows from Gronwall’s inequality.

• 2. For the second statement, using Ito’s formula and Assumption 2, it follows

  $$\begin{aligned}&\frac{d{{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4}{ds} \nonumber \\&\quad = \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x, \xi ^x_{\tau _2, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \nonumber \\&\qquad + 2\Vert \alpha _2(x,\xi ^x_{\tau _2, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) + 4\big |\big (\alpha _2(x,\xi ^x_{\tau _2, s})\\&\qquad - \alpha _2(x,\xi ^x_{\tau _1, s})\big )^T\big (\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}\big )\big |^2\Big ] \nonumber \\&\quad \le \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x, \xi ^x_{\tau _2, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \\&\qquad + 6\Vert \alpha _2(x,\xi ^x_{\tau _2, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) \Big ] \nonumber \\&\quad \le -\frac{4\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4\,. \end{aligned}$$

  Therefore, integrating and using the first statement above, we obtain

  $$\begin{aligned} {{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4 \le e^{-\frac{4\lambda (s-\tau _2)}{\epsilon }} {{\mathbf {E}}}|\xi ^x_{\tau _1, \tau _2} - y|^4 \le C\big (1+|x|^4 + |y|^4\big )e^{-\frac{4\lambda (s-\tau _2)}{\epsilon }}\,. \end{aligned}$$

• 3. For the third statement, in a similar way, applying Ito’s formula, using Assumption 2, as well as Lipschitz property of functions g and $\alpha _2$, we have

  $$\begin{aligned}&\frac{d{{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4}{ds} \nonumber \\&\quad = \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x', \xi ^{x'}_{\tau _2, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \nonumber \\&\qquad + 2\Vert \alpha _2(x',\xi ^{x'}_{\tau _2, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) + 4\big |\big (\alpha _2(x',\xi ^{x'}_{\tau _2, s})\\&\qquad - \alpha _2(x,\xi ^x_{\tau _1, s})\big )^T\big (\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}\big )\big |^2\Big ] \nonumber \\&\quad \le \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x', \xi ^{x'}_{\tau _2, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \\&\qquad + 6\Vert \alpha _2(x',\xi ^{x'}_{\tau _2, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) \Big ] \nonumber \\&\quad \le \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x', \xi ^{x'}_{\tau _2, s}) - g(x', \xi ^x_{\tau _1, s}), \xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \\&\qquad + 12\Vert \alpha _2(x',\xi ^{x'}_{\tau _2, s})-\alpha _2(x',\xi ^x_{\tau _1, s})\Vert ^2\big ) \Big ] \nonumber \\&\qquad + \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x', \xi ^{x}_{\tau _1, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \\&\qquad + 12\Vert \alpha _2(x',\xi ^{x}_{\tau _1, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) \Big ] \nonumber \\&\quad \le -\frac{4\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4\\&\qquad + \frac{C}{\epsilon } {{\mathbf {E}}}\big ( |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^3|x'-x|\big ) + \frac{C}{\epsilon } {{\mathbf {E}}}\big (|\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2|x'-x|^2\big ) \nonumber \\&\quad \le -\frac{2\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4 + \frac{C}{\epsilon } |x' - x|^4\,, \end{aligned}$$

  where inequality (5.9) is used to obtain the last inequality. Gronwall’s inequality together with the first statement above then yield the assertion.

$\square$

Now consider the derivative process

$$\begin{aligned} d\xi ^x_{\tau , s, x_i} =&\, \frac{1}{\epsilon } \Big (D_{x_i} g(x,\xi ^x_{\tau , s}) + \nabla _{y} g(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}\Big ) ds \\&+ \frac{1}{\sqrt{\epsilon }}\Big (D_{x_i} \alpha _2(x,\xi ^x_{\tau , s}) + \nabla _{y} \alpha _2(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}\Big ) dw_s\,, \end{aligned}$$

with $s\ge \tau \,, \xi ^x_{\tau , \tau ,x_i} = 0$, $1\le i \le k$. In the above, we used $D_{x_i}$ to denote derivatives with respect to scalar $x_i \in {\mathbb {R}}$ and $\nabla _y$ to denote derivatives with respect to a vector $y \in {\mathbb {R}}^l$. We summarize its properties in the following result.

Lemma 8.2

Under Assumptions 1–2, there exists a constant $C > 0$, independent of $\epsilon , x, y$, such that $\forall 1 \le i \le k$,

• 1. For $x \in {\mathbb {R}}^k$, $s \ge \tau$, ${{\mathbf {E}}}|\xi ^x_{\tau , s, x_i}|^4 \le C$.

• 2. For $\tau _1 \le \tau _2$, $x \in {\mathbb {R}}^k$,

  $$\begin{aligned} {{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2 \le C\big (1 + |x|^2+|y|^2\big ) e^{-\frac{\lambda (s-\tau _2)}{\epsilon }}\,. \end{aligned}$$

• 3. For $\tau _1 \le \tau _2$, $x, x' \in {\mathbb {R}}^k$,

  $$\begin{aligned} {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 \le Ce^{-\frac{\lambda (s-\tau _2)}{\epsilon }} \left[ 1 + \frac{s - \tau _2}{\epsilon } \big (1 + |x|^2+|y|^2\big )\right] + C |x - x'|^2\,. \end{aligned}$$

Proof

• 1. Using Ito’s formula, Assumption 1 (Lipschitz continuity of functions g and $\alpha _2$), inequality (3.11) in Remark 2, as well as inequality (5.9), we see that

  $$\begin{aligned} \frac{d{{\mathbf {E}}}|\xi ^x_{\tau , s, x_i}|^4}{ds} \le&\ \frac{1}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s, x_i}|^2\Big (4\langle D_{x_i} g(x,\xi ^x_{\tau , s}) + \nabla _{y} g(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}, \xi ^x_{\tau , s, x_i}\rangle \\&+ 6\Vert D_{x_i} \alpha _2(x,\xi ^x_{\tau , s}) + \nabla _{y} \alpha _2(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}\Vert ^2\Big )\Big ] \nonumber \\ \le&\frac{1}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s, x_i}|^2\Big (C|\xi ^x_{\tau , s, x_i}| + 4\langle \nabla _{y} g(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}, \xi ^x_{\tau , s, x_i}\rangle + C \\&+ 12\Vert \nabla _{y} \alpha _2(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}\Vert ^2\Big )\Big ] \nonumber \\ \le&-\frac{2\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau , s, x_i}|^4 + \frac{C}{\epsilon } \end{aligned}$$

  and therefore ${{\mathbf {E}}}|\xi ^x_{\tau , s, x_i}|^4 \le C$ by Gronwall’s inequality.

• 2. Now consider $\xi ^x_{\tau _1, s,x_i}, \xi ^x_{\tau _2, s,x_i}$ with $\tau _1 \le \tau _2$. Using Lipschitz condition of functions $g, \alpha _2$, inequality (3.11) in Remark 2, as well as inequality (5.9), it follows

  $$\begin{aligned}&\frac{d{{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2}{ds} \nonumber \\&\quad = \frac{2}{\epsilon } {{\mathbf {E}}}\langle D_{x_i} g(x, \xi ^x_{\tau _2, s}) - D_{x_i} g(x, \xi ^x_{\tau _1, s}) + \nabla _{y} g(x, \xi ^x_{\tau _2, s}) \xi ^x_{\tau _2, s, x_i} \\&\qquad - \nabla _{y} g(x, \xi ^x_{\tau _1, s}) \xi ^x_{\tau _1, s, x_i}, \xi ^x_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{1}{\epsilon } {{\mathbf {E}}}\Vert D_{x_i} \alpha _2(x, \xi ^x_{\tau _2, s}) - D_{x_i} \alpha _2(x, \xi ^x_{\tau _1, s}) + \nabla _{y} \alpha _2(x, \xi ^x_{\tau _2, s}) \xi ^x_{\tau _2, s, x_i} \\&\qquad - \nabla _{y} \alpha _2(x, \xi ^x_{\tau _1, s}) \xi ^x_{\tau _1, s, x_i}\Vert ^2 \nonumber \\&\quad \le \frac{C}{\epsilon }{{\mathbf {E}}}\Big (|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}||\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|\Big ) + \frac{2}{\epsilon } {{\mathbf {E}}}\langle \big (\nabla _{y} g(x, \xi ^x_{\tau _2, s})\\&\qquad -\nabla _{y} g(x, \xi ^x_{\tau _1, s})\big )\xi ^x_{\tau _1, s, x_i}, \xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{2}{\epsilon } {{\mathbf {E}}}\langle \nabla _{y} g(x, \xi ^x_{\tau _2, s})(\xi ^x_{\tau _2, s, x_i} {-} \xi ^x_{\tau _1, s, x_i}), \xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}\rangle {+} \frac{C}{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2\nonumber \\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert \big (\nabla _{y} \alpha _2(x, \xi ^x_{\tau _2, s}){-}\nabla _{y} \alpha _2(x, \xi ^x_{\tau _1, s})\big ) \xi ^x_{\tau _1, s, x_i}\Vert ^2\\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert \nabla _{y} \alpha _2(x, \xi ^x_{\tau _2, s})(\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i})\Vert ^2 \nonumber \\&\quad \le -\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon } \big ({{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4\big )^{\frac{1}{2}} \big ({{\mathbf {E}}}|\xi ^x_{\tau _1, s, x_i}|^4)^{\frac{1}{2}} \\&\qquad + \frac{C}{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \nonumber \\&\quad \le -\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon } (1 + |x|^2+|y|^2) e^{-\frac{2\lambda (s-\tau _2)}{\epsilon }}\,, \end{aligned}$$

  where the first assertion above and Lemma 8.1 have been used in the last inequality. Then Gronwall’s inequality entails

  $$\begin{aligned} {{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2 \le C\big (1 + |x|^2+|y|^2\big ) e^{-\frac{\lambda (s-\tau _2)}{\epsilon }}\,. \end{aligned}$$

• 3. Consider $\xi ^{x}_{\tau _1, s,x_i}, \xi ^{x'}_{\tau _2, s,x_i}$ with $\tau _1 \le \tau _2$. In a similar way, we have

  $$\begin{aligned}&\frac{d{{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2}{ds} \nonumber \\&\quad = \frac{2}{\epsilon } {{\mathbf {E}}}\langle D_{x_i} g(x', \xi ^{x'}_{\tau _2, s}) - D_{x_i} g(x, \xi ^x_{\tau _1, s}) + \nabla _{y} g(x', \xi ^{x'}_{\tau _2, s}) \xi ^{x'}_{\tau _2, s, x_i} \\&\qquad - \nabla _{y} g(x, \xi ^x_{\tau _1, s}) \xi ^x_{\tau _1, s, x_i}, \xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{1}{\epsilon } {{\mathbf {E}}}\Vert D_{x_i} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) - D_{x_i} \alpha _2(x, \xi ^x_{\tau _1, s}) + \nabla _{y} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) \xi ^{x'}_{\tau _2, s, x_i} \\&\qquad - \nabla _{y} \alpha _2(x, \xi ^x_{\tau _1, s}) \xi ^x_{\tau _1, s, x_i}\Vert ^2 \nonumber \\&\quad \le \frac{2}{\epsilon } {{\mathbf {E}}}\langle D_{x_i} g(x', \xi ^{x'}_{\tau _2, s}) - D_{x_i} g(x', \xi ^x_{\tau _1, s}) + \nabla _{y} g(x', \xi ^{x'}_{\tau _2, s})\\&\qquad (\xi ^{x'}_{\tau _2, s, x_i} -\xi ^x_{\tau _1, s, x_i}), \xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{2}{\epsilon } {{\mathbf {E}}}\langle D_{x_i} g(x', \xi ^{x}_{\tau _1, s}) - D_{x_i} g(x, \xi ^x_{\tau _1, s}) + \big (\nabla _{y} g(x', \xi ^{x'}_{\tau _2, s}) \\&\qquad - \nabla _{y} g(x, \xi ^{x}_{\tau _1, s})\big ) \xi ^x_{\tau _1, s, x_i}, \xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert D_{x_i} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) - D_{x_i} \alpha _2(x, \xi ^x_{\tau _1, s})\Vert ^2\\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert \nabla _{y} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) (\xi ^{x'}_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i})\Vert ^2 \nonumber \\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert \big (\nabla _{y} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) - \nabla _{y} \alpha _2(x, \xi ^x_{\tau _1, s})\big ) \xi ^x_{\tau _1, s, x_i}\Vert ^2 \nonumber \\&\quad \le -\frac{2\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon } {{\mathbf {E}}}\big (|\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}||\xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}|\big )\\&\qquad + \frac{C}{\epsilon } {{\mathbf {E}}}\big (|x' - x||\xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}|\big ) \nonumber \\&\qquad +\frac{C}{\epsilon } {{\mathbf {E}}}\big [\big (|x'-x|+ |\xi ^{x'}_{\tau _2, s}-\xi ^x_{\tau _1, s}|\big )|\xi ^x_{\tau _1, s, x_i}||\xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}|\big ]\\&\qquad + \frac{C}{\epsilon } |x-x'|^2+\frac{C}{\epsilon }{{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s}-\xi ^x_{\tau _1, s}|^2 \nonumber \\&\qquad + \frac{C}{\epsilon } {{\mathbf {E}}}\big [(|x'-x| + |\xi ^{x'}_{\tau _2, s}-\xi ^x_{\tau _1, s}|\big )^2|\xi ^x_{\tau _1, s, x_i}|^2\big ] \nonumber \\&\quad \le -\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon }\Big (|x' - x|^2 + {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^2 \\&\qquad + ({{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4)^{\frac{1}{2}}\Big ) \nonumber \\&\quad \le -\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon } \Big [(1 + |x|^2+|y|^2) e^{-\frac{\lambda (s-\tau _2)}{\epsilon }} + |x' - x|^2\Big ]\,, \end{aligned}$$

  and thus

  $$\begin{aligned} {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 \le Ce^{-\frac{\lambda (s-\tau _2)}{\epsilon }} \Big [ 1 + \frac{s - \tau _2}{\epsilon } (1 + |x|^2+|y|^2)\Big ] + C |x' - x|^2\,. \end{aligned}$$

$\square$

The above results allow us to define the stationary process $\xi _s^x=\xi _{-\infty ,s}^{x}$ with $\xi _{s}^{x}\sim \rho _{x}(y)\,dy$ where $\rho _{x}$ is the stationary probability density with respect to Lebesgue measure, and also the derivative process $\xi ^x_{s, x_i}$ for $1 \le i \le k$, satisfying that $\forall f \in C^1_b({\mathbb {R}}^k\times {\mathbb {R}}^l)$ and ${\widetilde{f}}(x) = {{\mathbf {E}}}(f(x, \xi _s^x)) = \int _{{\mathbb {R}}^l} f(x,y) \rho _x(y) dy$, it holds

$$\begin{aligned} D_{x_i} {\widetilde{f}}(x) = {{\mathbf {E}}}\big (D_{x_i}f(x, \xi _s^x) + \nabla _{y}f(x,\xi _s^x)\xi ^x_{s,x_i}\big )\,. \end{aligned}$$

(8.2)

The processes $\xi ^x_s$ and $\xi ^x_{s, x_i}$ have the following properties:

Lemma 8.3

Under Assumptions 1 and 2, there is a constant $C>0$, independent of $\epsilon$, x and y, such that $\forall f \in C_b^1({\mathbb {R}}^l)$:

• 1.
  $$\begin{aligned} \Big | {{\mathbf {E}}}f(\xi ^x_{0,s}) - \int _{{\mathbb {R}}^l} f(y) \rho _x(y) dy\Big | \le \sup |f'| \Big (|x| + |y| + 1\Big ) e^{-\frac{\lambda s}{\epsilon }}\,. \end{aligned}$$

  (8.3)
• 2.
  $$\begin{aligned}&\Big | {{\mathbf {E}}}\Big (f(\xi ^x_{0,s})\xi ^x_{0,s,x_i}\Big ) -{{\mathbf {E}}}\Big (f(\xi ^x_{s})\xi ^x_{s,x_i}\Big ) \Big |\nonumber \\&\quad \le C\Big (\sup |f| + \sup |f'|\Big ) \Big (1 + |x| + |y|\Big ) e^{-\frac{\lambda s}{2\epsilon }}\,. \end{aligned}$$

  (8.4)

Proof

We only prove the second inequality, as the first one follows in a similar fashion. Using Lemmas 8.1 and 8.2, we readily conclude that

$$\begin{aligned}&\Big | {{\mathbf {E}}}\big (f(\xi ^x_{0,s})\xi ^x_{0,s,x_i}\big ) -{{\mathbf {E}}}\big (f(\xi ^x_{s})\xi ^x_{s,x_i}\big ) \Big | \nonumber \\&\quad \le \Big | {{\mathbf {E}}}\big [f(\xi ^x_{s})(\xi ^x_{0,s,x_i} -\xi ^x_{s,x_i})\big ]\Big | + \Big |{{\mathbf {E}}}\big [(f(\xi ^x_{0,s}) - f(\xi ^x_{s}))\xi ^x_{0, s,x_i}\big ] \Big | \nonumber \\&\quad \le C\big (\sup |f| + \sup |f'|\big ) \big (1 + |x| + |y|\big ) e^{-\frac{\lambda s}{2\epsilon }} \end{aligned}$$

$\square$

An analogous property for the stationary process $\xi ^x_{s}$ is the following:

Lemma 8.4

Under Assumption 1 and 2, there exists constant $C > 0$, independent of $x, x'$, such that

• 1. For $x \in {\mathbb {R}}^k$, ${{\mathbf {E}}}|\xi ^x_{s, x_i}|^4 \le C$.

• 2. For $x , x'\in {\mathbb {R}}^k$, ${{\mathbf {E}}}|\xi ^{x'}_{s} - \xi ^x_{s}|^4 \le C |x - x'|^4$.

• 3. For $x, x' \in {\mathbb {R}}^k$, ${{\mathbf {E}}}|\xi ^{x'}_{s, x_i} - \xi ^{x}_{s, x_i}|^2 \le C |x - x'|^2$.

Proof

The conclusions follow directly by letting $\tau _1, \tau _2\rightarrow -\infty$ in Lemma 8.1 and 8.2. $\square$