Asymptotic Behavior for Multi-scale SDEs with Monotonicity Coefficients Driven by Lévy Processes

Abstract

In this paper, we study the asymptotic behavior for multi-scale stochastic differential equations driven by Lévy processes. The optimal strong convergence order 1/2 is obtained by studying the regularity estimates for the solution of Poisson equation with polynomial growth coefficients, and the optimal weak convergence order 1 is got by using the technique of Kolmogorov equation. The main contribution is that the obtained results can be applied to a class of multi-scale stochastic differential equations with monotonicity coefficients, as well as the driven processes can be the general Lévy processes, which seems new in the existing literature.

Appendix

1.1 Differentiability of $Y^{x,y}$

In this subsection, we give the proofs of the differentiability of the solution $Y^{x,y}_t$ of the frozen equation with respect to parameter x and y.

Proposition 6.1

Under the assumptions B1-B3. Then $Y^{x,y}_t$ is differentiable with respect to y and x in directions $l\in \mathbb {R}^m$ and $l_1\in \mathbb {R}^n$ respectively, which satisfies

$$\begin{aligned} d[\partial _yY_{t}^{x,y}\cdot l]= & {} \partial _{y} f(x,Y_{t}^{x,y})\cdot (\partial _yY_{t}^{x,y}\cdot l)dt+\partial _yg(x,Y_{t}^{x,y})\cdot (\partial _yY_{t}^{x,y}\cdot l)d\tilde{W}_{t}^{2} \nonumber \\+ & {} \int _{\mathcal {Z}_2}\partial _yh_2(x,Y_{t-}^{x,y},z) \cdot (\partial _yY_{t-}^{x,y}\cdot l)\tilde{N}^{2}(dz,dt). \end{aligned}$$

(1.1)

and

$$\begin{aligned} d\left[ \partial _xY_{t}^{x,y}\cdot l_1\right]= & {} \left[ \partial _x f(x,Y_{t}^{x,y})\cdot l_1+\partial _yf(x,Y_{t}^{x,y})\cdot (\partial _xY_{t}^{x,y}\cdot l_1)\right] dt\nonumber \\+ & {} \left[ \partial _x g(x,Y_{t}^{x,y})\cdot l_1+\partial _yg(x,Y_{t}^{x,y})\cdot ( \partial _xY_{t}^{x,y}\cdot l_1)\right] d \tilde{W}_{t}^{2} \nonumber \\+ & {} \int _{\mathcal {Z}_2}\left[ \partial _x h_2(x,Y_{t-}^{x,y},z)\cdot l_1\!+\!\partial _y h_2(x,Y_{t-}^{x,y},z) \cdot (\partial _x Y_{t-}^{x,y}\cdot l_1)\right] \nonumber \\{} & {} \times \tilde{N}^{2}(dt,dz). \end{aligned}$$

(1.2)

Moreover, there exist $C,\gamma >0$ such that

$$\begin{aligned} \mathbb {E}|\partial _yY_{t}^{x,y}\cdot l|^{\ell }\leqslant 2 e^{-\gamma t}|l|^{\ell },\quad \sup _{t\ge 0}\mathbb {E}|\partial _x Y^{x,y}_t\cdot l_1|^{\ell } \le C|l_1|^{\ell }, \end{aligned}$$

(1.3)

where $\ell$ is the constant in assumption B1.

Proof

We only prove $Y^{x,y}_t$ is differentiable with respect to y in direction $l\in \mathbb {R}^m$ and its directional derivative $\partial _yY^{x,y}_t\cdot l$ satisfies equation (6.1). Since $Y^{x,y}_t$ is differentiable with respect to x in direction $l_1\in \mathbb {R}^n$ and its directional derivative $\partial _xY^{x,y}_t\cdot l_1$ satisfies equation (6.2) can be proved by a similar argument, thus we omit the details.

In fact, it is sufficient to prove the following result:

$$\begin{aligned} \lim _{\delta \rightarrow 0}\mathbb {E}\left| \frac{Y^{x,y+\delta l}_t-Y^{x,y}_t}{\delta }-\partial _y Y^{x,y}_t\cdot l\right| ^2=0. \end{aligned}$$

(1.4)

To do this, denote $Z^{\delta ,l}_t:=\frac{Y^{x,y+\delta l}_t-Y^{x,y}_t}{\delta }-\partial _yY_{t}^{x,y}\cdot l$, recall

$$\begin{aligned}\left\{ \begin{array}{l} \displaystyle dY_{t}^{x,y}=f(x,Y_{t}^{x,y})dt+g(x,Y_{t}^{x,y})d \tilde{W}_{t}^{2}+\int _{\mathcal {Z}_2}h_{2}(x,Y_{t-}^{x,y},z)\tilde{N}^{2}(dz,dt),\\ Y_{0}^{x,y}=y,\\ \end{array}\right. \end{aligned}$$

then $Z^{\delta ,l}_t$ satisfies the following equation:

$$\begin{aligned}\left\{ \begin{array}{l} \displaystyle dZ^{\delta ,l}_t=\left[ \frac{f(x,Y_{t}^{x,y+\delta l})-f(x,Y_{t}^{x,y})}{\delta }-\partial _yf(x,Y_{t}^{x,y})\cdot (\partial _yY_{t}^{x,y}\cdot l)\right] dt\\ \quad \quad \quad \quad +\left[ \frac{g(x,Y_{t}^{x,y+\delta l})-g(x,Y_{t}^{x,y})}{\delta }-\partial _yg(x,Y_{t}^{x,y})\cdot (\partial _yY_{t}^{x,y}\cdot l)\right] d\tilde{W}_{t}^{2}\\ \quad \quad \quad \quad +\int _{\mathcal {Z}_2}\left[ \frac{h_2(x,Y_{t-}^{x,y+\delta l},z)-h_2(x,Y_{t-}^{x,y},z)}{\delta }-\partial _yh_2(x,Y_{t-}^{x,y},z)\cdot (\partial _yY_{t-}^{x,y}\cdot l)\right] \tilde{N}^{2}(dz,dt),\\ Z^{\delta ,l}_0=0\\ \end{array}\right. \end{aligned}$$

By Itô’s formula and taking expectation, we have

$$\begin{aligned} \frac{d}{dt}\mathbb {E}|Z^{\delta ,l}_t|^2= & {} 2\mathbb {E}\left\langle \left[ \frac{f(x,Y_{t}^{x,y+\delta l})-f(x,Y_{t}^{x,y})}{\delta }-\partial _yf(x,Y_{t}^{x,y})\cdot (\partial _yY_{t}^{x,y}\cdot l)\right] , Z^{\delta ,l}_t\right\rangle \nonumber \\{} & {} +\mathbb {E}\left\| \frac{g(x,Y_{t}^{x,y+\delta l})-g(x,Y_{t}^{x,y})}{\delta }-\partial _yg(x,Y_{t}^{x,y})\cdot (\partial _yY_{t}^{x,y}\cdot l)\right\| ^2\nonumber \\{} & {} +\mathbb {E}\int _{\mathcal {Z}_2}\left| \frac{h_2(x,Y_{t}^{x,y+\delta l},{z})-h_2(x,Y_{t}^{x,y},{z})}{\delta }-\partial _yh_2(x,Y_{t}^{x,y},{z})\cdot (\partial _yY_{t}^{x,y}\cdot l)\right| ^2\nu _2(dz)\nonumber \\ =:{} & {} \sum ^{3}_{i=1}\tilde{I}_i(t). \end{aligned}$$

(1.5)

For the term $\tilde{I}_1(t)$. By Taylor’s formula, there exists $\xi \in (0,1)$ such that

$$\begin{aligned} \tilde{I}_1(t)= & {} 2\mathbb {E}\left\langle \left[ \frac{f(x,Y_{t}^{x,y+\delta l})-f(x,Y_{t}^{x,y})}{\delta }-\partial _yf(x,Y_{t}^{x,y})\cdot \frac{Y^{x,y+\delta l}_t-Y^{x,y}_t}{\delta }\right] , Z^{\delta ,l}_t\right\rangle \nonumber \\{} & {} +2\mathbb {E}\left\langle \partial _yf(x,Y_{t}^{x,y})\cdot Z^{\delta ,l}_t, Z^{\delta ,l}_t\right\rangle \nonumber \\\leqslant & {} 2\mathbb {E}\delta ^{-1} \Vert \partial ^2_yf(x,\xi Y_{t}^{x,y+\delta l}\!\!+(1-\xi )Y_{t}^{x,y})\Vert |Y_{t}^{x,y+\delta l}-Y_{t}^{x,y}|^2| Z^{\delta ,l}_t| \nonumber \\{} & {} +2\mathbb {E}\left\langle \partial _yf(x,Y_{t}^{x,y})\cdot Z^{\delta ,l}_t, Z^{\delta ,l}_t\right\rangle . \end{aligned}$$

(2.1)

For the terms $\tilde{I}_2(t)$ and $\tilde{I}_3(t)$. By a similar argument above, there exist $\xi _2,\xi _3\in (0,1)$ such that

$$\begin{aligned} \tilde{I}_2(t)\leqslant & {} 2 2\mathbb {E}\left\| \frac{g(x,Y_{t}^{x,y+\delta l})-g(x,Y_{t}^{x,y})}{\delta }-\partial _yg(x,Y_{t}^{x,y})\cdot \frac{Y^{x,y+\delta l}_t-Y^{x,y}_t}{\delta }\right\| ^2\nonumber \\{} & {} +2\mathbb {E}\left\| \partial _yg(x,Y_{t}^{x,y})\cdot Z^{\delta ,l}_t\right\| ^2\nonumber \\\leqslant & {} 2 2^{-1}\mathbb {E}\delta ^{-2} \Vert \partial ^2_yg(x,\xi _2 Y_{t}^{x,y+\delta l}\!\!+(1-\xi _2)Y_{t}^{x,y})\Vert ^2 |Y_{t}^{x,y+\delta l}-Y_{t}^{x,y}|^4\nonumber \\{} & {} +2\mathbb {E}\left\| \partial _yg(x,Y_{t}^{x,y})\cdot Z^{\delta ,l}_t\right\| ^2 \end{aligned}$$

(2.2)

and

$$\begin{aligned} \tilde{I}_3(t)\leqslant & {} 2 2\mathbb {E}\int _{\mathcal {Z}_2}\left| \frac{h_2(x,Y_{t}^{x,y+\delta l},z)-h_2(x,Y_{t}^{x,y},z)}{\delta }-\partial _yh_2(x,Y_{t}^{x,y},z)\cdot \frac{Y^{x,y+\delta l}_t-Y^{x,y}_t}{\delta }\right| ^2\nu _2(dz)\nonumber \\{} & {} +2\mathbb {E}\int _{\mathcal {Z}_2}\left| \partial _yh_2(x,Y_{t}^{x,y},z)\cdot Z^{\delta ,l}_t\right| ^2\nu _2(dz)\nonumber \\\leqslant & {} 2 2^{-1}\mathbb {E}\int _{\mathcal {Z}_2}\delta ^{-2} \Vert \partial ^2_yh_2(x,\xi _3 Y_{t}^{x,y+\delta l}\!\!+(1-\xi _3)Y_{t}^{x,y},z)\Vert ^2 |Y_{t}^{x,y+\delta l}-Y_{t}^{x,y}|^4\nu _2(dz)\nonumber \\{} & {} +2\mathbb {E}\int _{\mathcal {Z}_2}\left| \partial _yh_2(x,Y_{t}^{x,y},z)\cdot Z^{\delta ,l}_t\right| ^2\nu _2(dz). \end{aligned}$$

(2.3)

Combining (6.5)-(6.8), then by Young’s inequality, assumption B3, (3.18) and Lemma 3.3, there exists $k>0$ such that

$$\begin{aligned} \frac{d}{dt}\mathbb {E}|Z^{\delta ,l}_t|^2&\leqslant 2&\frac{\beta }{2}\mathbb {E}|Z^{\delta ,l}_t|^2+C(1+|x|^k+|y|^k)\delta ^2\\{} & {} +2\mathbb {E}\left\langle \partial _yf(x,Y_{t}^{x,y})\cdot Z^{\delta ,l}_t, Z^{\delta ,l}_t\right\rangle +2\mathbb {E}\left\| \partial _yg(x,Y_{t}^{x,y})\cdot Z^{\delta ,l}_t\right\| ^2\\{} & {} +2\mathbb {E}\int _{\mathcal {Z}_2}\left| \partial _yh_2(x,Y_{t}^{x,y},z)\cdot Z^{\delta ,l}_t\right| ^2\nu _2(dz). \end{aligned}$$

Note that condition (2.8) implies that for any $x\in \mathbb {R}^n,y\in \mathbb {R}^m, l\in \mathbb {R}^m$

$$\begin{aligned}{} & {} 2\left\langle \partial _yf(x,y)\cdot l,l\right\rangle +(\ell -1)\Vert \partial _yg(x,y)\cdot l\Vert ^{2}\nonumber \\{} & {} \quad +2^{\ell -3}(\ell -1)\!\int _{\mathcal {Z}_2}\!\!|\partial _yh_{2}(x,y,z)\cdot l|^{2}\nu _2(dz)\le -\beta |l|^{2}, \end{aligned}$$

(2.4)

where $\ell >8$, thus this implies that

$$\begin{aligned}{} & {} 2\left\langle \partial _yf(x,y)\cdot l,l\right\rangle +2\Vert \partial _yg(x,y)\cdot l\Vert ^{2}\\{} & {} \quad +2\int _{\mathcal {Z}_2}\!\!|\partial _yh_{2}(x,y,z)\cdot l|^{2}\nu _2(dz)\le -\beta |l|^{2}. \end{aligned}$$

Then we have

$$\begin{aligned} \frac{d}{dt}\mathbb {E}|Z^{\delta ,l}_t|^2\leqslant 2 -\frac{\beta }{2}\mathbb {E}|Z^{\delta ,l}_t|^2+C(1+|x|^k+|y|^k)\delta ^2. \end{aligned}$$

By comparison theorem, we have

$$\begin{aligned} \mathbb {E}|Z^{\delta ,l}_t|^2\leqslant 2 C_{\beta }(1+|x|^k+|y|^k)\delta ^2\rightarrow 0,\quad \text {as} \quad \delta \rightarrow 0. \end{aligned}$$

Using Itô’s formula on $|\partial _yY_{t}^{x,y}\cdot l|^{\ell }$ and taking expectation on both sides, then by (3.4) again, we have

$$\begin{aligned} \frac{d}{dt}\mathbb {E}|\partial _yY_{t}^{x,y}\cdot l|^{\ell }\leqslant & {} 2{\ell }\mathbb {E}\left[ \big |\partial _yY_{t}^{x,y}\cdot l\big |^{\ell -2}\left\langle \partial _yf(x,Y_{t}^{x,y})\cdot (\partial _yY_{t}^{x,y}\cdot l), \partial _yY_{t}^{x,y}\cdot l\right\rangle \right] \\{} & {} +\frac{{\ell }({\ell }-1)}{2}\mathbb {E}\left[ \big |\partial _yY_{t}^{x,y}\cdot l\big |^{\ell -2}\left\| \partial _yg(x,Y_{t}^{x,y})\cdot (\partial _yY_{t}^{x,y}\cdot l)\right\| ^2\right] \\{} & {} +2^{\ell -4}\ell (\ell -1)\mathbb {E}\Big [|\partial _yY_{t}^{x,y}\cdot l|^{\ell -2} \int _{\mathcal {Z}_2}|\partial _yh_2(x,Y_{t-}^{x,y},z) \cdot (\partial _yY_{t-}^{x,y}\cdot l)|^2\nu _2(dz)\Big ]\\{} & {} +2^{\ell -4}\ell (\ell -1)\mathbb {E} \int _{\mathcal {Z}_2}|\partial _yh_2(x,Y_{t-}^{x,y},z) \cdot (\partial _yY_{t-}^{x,y}\cdot l)|^{\ell }\nu _2(dz). \end{aligned}$$

Note that condition (2.9) implies that for any $x\in \mathbb {R}^n,y\in \mathbb {R}^m, l\in \mathbb {R}^m$

$$\begin{aligned} 2^{\ell -3}(\ell -1)\!\int _{\mathcal {Z}_2}\!\!|\partial _yh_{2}(x,y,z)\cdot l|^{\ell }\nu _2(dz)\le L_{h_2}|l|^{\ell }. \end{aligned}$$

(2.5)

By (6.9) and (6.10), we have

$$\begin{aligned} \frac{d}{dt}\mathbb {E}|\partial _yY_{t}^{x,y}\cdot l|^{\ell } \le{} & {} -\frac{\ell \beta }{2}\mathbb {E}\left[ \big |\partial _yY_{t}^{x,y}\cdot l\big |^{\ell }\right] +\frac{\ell L_{h_2}}{2}\mathbb {E}\big |\partial _yY_{t}^{x,y}\cdot l\big |^{\ell }\\ ={} & {} -\frac{\ell (\beta -L_{h_2})}{2}\mathbb {E}\big |\partial _yY_{t}^{x,y}\cdot l\big |^{\ell }. \end{aligned}$$

Then by the comparison theorem, we get

$$\begin{aligned} \mathbb {E}|\partial _yY_{t}^{x,y}\cdot l|^{\ell }\le e^{-\frac{\ell (\beta -L_{h_2})t}{2}}|l|^{\ell }. \end{aligned}$$

Thus the first estimate in (6.3) holds. By a similar argument, we can prove the second estimate in (6.3). The proof is complete. $\square$

Remark 6.2

It is worth noting that by a similar argument to that above, using the additional regularity assumptions (2.11) in B3 on the coefficients, we can further prove the differentiability of $\partial _yY_{t}^{x,y}\cdot l$ and $\partial _xY_{t}^{x,y}\cdot l_1$ with respect to parameters. Let $\partial _y\partial _x Y^{x,y}_t\cdot (l_1,l_2)$ be the directional derivative of $\partial _x Y^{x,y}_t\cdot l_1$ with respect to y in the direction $l_2$. Let $\partial ^2_x Y^{x,y}_t\cdot (l_1,l_2)$ be the directional derivative of $\partial _x Y^{x,y}_t\cdot l_1$ with respect to x in the direction $l_2$. Let $\partial _y\partial _{x}^{2}Y_{t}^{x,y}\cdot (l_1,l_2,l_3)$ be the directional derivative of $\partial ^2_x Y^{x,y}_t\cdot (l_1,l_2)$ with respect to y in the direction $l_3$. We can easily prove for any unit vectors $l_1,l_2,l_3$,

$$\begin{aligned}{} & {} \mathbb {E}|\partial _y\partial _{x}Y^{x,y}_t\cdot (l_1,l_2)|^4\le Ce^{-4\gamma t}(1+|y|^{4k}),\\{} & {} \sup _{t\ge 0}\mathbb {E}|\partial ^2_{x}Y^{x,y}_t\cdot (l_1,l_2)|^4\le C(1+|y|^{4k}),\\{} & {} \mathbb {E}|\partial _y\partial ^2_{x}Y^{x,y}_t\cdot (l_1,l_2,l_3)|^2\le Ce^{-2\gamma t}(1+|y|^{4k}), \end{aligned}$$

where $C,k,\gamma >0$.

1.2 Well-posedness of equation (5.20)

In this subsection, we give the detailed proof of the existence and uniqueness of equation (5.20).

Proposition 6.3

Under the assumptions A1-A3 and B1-B3. For any $\phi \in C^{2}_p(\mathbb {R}^{n})$, the following Kolmogorov equation

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _t u(t,x)=\bar{\mathscr {L}}_1 u(t,x),\quad t\ge 0,x\in \mathbb {R}^n, \\ u(0, x)=\phi (x), \end{array}\right. \end{aligned}$$

(2.6)

admits a unique solution $u\in C^{1,2}(\mathbb {R}_{+}\times \mathbb {R}^n)$, moreover the solution u is given by

$$ u(t,x)=\mathbb {E}\phi (\bar{X}^x_t), $$

where $\bar{\mathscr {L}}_1$ is the infinitesimal generator of the transition semigroup of the averaged equation (5.14), which is given by

$$\begin{aligned}{} & {} \bar{\mathscr {L}}_1\phi (x):=\langle \bar{b}(x), D \phi (x)\rangle +\frac{1}{2}\text {Tr}[\bar{\sigma }\bar{\sigma }(x)D^{2}\phi (x)]\\{} & {} \quad \quad \quad \quad \quad \quad +\int _{\mathcal {Z}_1}\left[ \phi (x+h_{1}(x,z))\!-\!\phi (x)\!-\!\langle D \phi (x),h_{1}(x,z)\rangle \right] \nu _1(dz). \end{aligned}$$

Proof

Existence: Using Itô’s formula, it is easy to see $u(t,x)=\mathbb {E}\phi (\bar{X}^x_t)$ is differentiable with respect to t. Moreover, using the chain rule and $\phi \in C^{2}_p(\mathbb {R}^{n})$, it is easy to see u(t, x) is first and second differentiable with respect to x. Hence $u\in C^{1,2}(\mathbb {R}_{+}\times \mathbb {R}^n)$. In order to prove u(t, x) solves equation (6.11), we use the definition of generator $\bar{\mathscr {L}}_1$, more precisely, by the Markov property and homogeneous property, we have for any $s>0$,

$$\begin{aligned} \frac{\mathbb {E}u(t,\bar{X}^x_s)-u(t,x)}{s}= & {} \frac{\mathbb {E}\left[ \mathbb {E}\phi (\bar{X}^{y}_{t})|_{y=\bar{X}^x_s}\right] -\mathbb {E}\phi (\bar{X}^x_t)}{s}\\= & {} \frac{\mathbb {E}\left[ \mathbb {E}\phi (\bar{X}^x_{t+s})| \mathscr {F}_{s}\right] -\mathbb {E}\phi (\bar{X}^x_t)}{s}\\= & {} \frac{\mathbb {E}\phi (\bar{X}^x_{t+s})-\mathbb {E}\phi (\bar{X}^x_t)}{s}\\= & {} \frac{u(t+s,x)-u(t,x)}{s}. \end{aligned}$$

Then letting $s\rightarrow 0$, we get (6.11).

Uniqueness: Let $w(t,x)\in C^{1,2}(\mathbb {R}_{+}\times \mathbb {R}^n)$ be another solution of (6.11) with $w(0,x)=\phi (x)$. For any fixed $t>0$, define

$$ \tilde{w}(s,x)=w(t-s,x), s\in [ 0,t], $$

then it is easy to check

$$ \partial _s \tilde{w}(s,x)+\bar{\mathscr {L}}_1 \tilde{w}(s,x)=0,\quad \forall s>0, x\in \mathbb {R}^n. $$

Then using Itô’s formula on $\tilde{w}(t, \bar{X}^{x}_t)$ and taking expectation, we have

$$\begin{aligned} \mathbb {E}\tilde{w}(t, \bar{X}^{x}_t)= & {} \tilde{w}(0, x)+\int ^t_0 \left[ \partial _s \tilde{w}(s,\bar{X}^{x}_s)+\bar{\mathscr {L}}_1 \tilde{w}(s,\bar{X}^{x}_s)\right] ds=\tilde{w}(0, x). \end{aligned}$$

Note that by the definition of $\tilde{w}$, it follows

$$ \mathbb {E}\tilde{w}(t, \bar{X}^{x}_t)=\mathbb {E}\phi (\bar{X}^{x}_t),\quad \tilde{w}(0, x)=w(t,x). $$

Hence, we obtain $w(t,x)=u(t,x)$. $\square$