General Large Deviations and Functional Iterated Logarithm Law for Multivalued Stochastic Differential Equations

Abstract

In this paper, we prove a large deviation principle of Freidlin–Wentzell type for multivalued stochastic differential equations (MSDEs) that is a little more general than the results obatined by Ren et al. (J Theor Prob 23:1142–1156, 2010). As an application, we derive a functional iterated logarithm law for the solutions of MSDEs.

Appendix: Proof of (9)

Lemma 6.1

Assume that (H1)–(H4) hold. Let $X^{\epsilon ,h_{\epsilon }}$ and $\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha }$ solve Eqs. (4) and (8), respectively. Then for any $x\in \overline{D(A)}$, we have

$$\begin{aligned} \lim _{\alpha \rightarrow 0}\mathbf{E}\sup _{t\in [0,T]}|\widetilde{X}^{\epsilon , h_{\epsilon ,\alpha }}(t)-X^{\epsilon ,h_{\epsilon }}(t)|^2=0 \end{aligned}$$

uniformly in $\epsilon >0$.

Proof

We denote by $M_{\epsilon }^n(t)$ the $\mathcal {C}^{\infty }$-approximation of $M_{\epsilon }(t)$, which is defined as follows:

$$\begin{aligned} M_{\epsilon }^n(t):=n\int \limits _{t-\frac{1}{n}}^{t+\frac{1}{n}}M_{\epsilon }\left( s-\frac{1}{n}\right) \rho (n(t-s)){\mathord {\mathrm{d}}}s, \end{aligned}$$

where $\rho$ is a mollifier with $supp \rho \subset (-1,1)$, $\rho \in C^{\infty }$ and $\int \nolimits _{-1}^1\rho (s){\mathord {\mathrm{d}}}s=1$. It is easy to obtain from (6) and (7)

$$\begin{aligned}&M_{\epsilon }^n(0)=0,\\&\lim _{n\rightarrow \infty }\mathbf{E}\sup _{t\in [0,T]}|M_{\epsilon }^n(t)-M_{\epsilon }(t)|^2=0,\\&\sup _{t\in [0,T]}|M_{\epsilon }^n(t)|\leqslant \sup _{t\in [0,T]}|M_{\epsilon }(t)|\quad \text {a.s.},\\&\sup _{|t'-t|\leqslant \delta }|M_{\epsilon }^n(t')-M_{\epsilon }^n(t)|\leqslant \sup _{|t'-t|\leqslant \delta }|M_{\epsilon }(t')-M_{\epsilon }(t)|\quad \text {a.s.},\forall \delta >0,\\&\mathbf{E}\sup _{t\in [0,T]}|\dot{M}_{\epsilon }^n(t)|^2+\mathbf{E}\sup _{t\in [0,T]}|\ddot{M}_{\epsilon }^n(t)|^2\leqslant C_n, \end{aligned}$$

where $C_n$ is a constant which does not depend on $\epsilon$ and $\alpha$.

Let $(X^{\epsilon ,h_{\epsilon },n}(\cdot ,x),K^{\epsilon ,h_{\epsilon },n}(\cdot ,x))$ solve the following MSDE:

$$\begin{aligned} \left\{ \begin{array}{l} {\mathord {\mathrm{d}}}X^{\epsilon ,h_{\epsilon },n}(t)\in {\mathord {\mathrm{d}}}M_{\epsilon }^n(t)-A_{\epsilon }(X^{\epsilon ,h_{\epsilon },n}(t)){\mathord {\mathrm{d}}}t,\\ X^{\epsilon ,h_{\epsilon },n}(0)=x, \end{array}\right. \end{aligned}$$

and $\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t,x)$ solve the following differential equation:

$$\begin{aligned} \left\{ \begin{array}{l} {\mathord {\mathrm{d}}}\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t)={\mathord {\mathrm{d}}}M_{\epsilon }^n(t)-A_{\epsilon }^{\alpha }(\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t)){\mathord {\mathrm{d}}}t,\\ \widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(0)=x. \end{array}\right. \end{aligned}$$

Now we split our proof into three steps.

Step 1

First we prove

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbf{E}\sup _{t\in [0,T]}|X^{\epsilon ,h_{\epsilon },n}(t)-X^{\epsilon ,h_{\epsilon }}(t)|=0 \end{aligned}$$

(23)

uniformly in $\epsilon$. For this, we proceed as follows. By [10, Proposition 4.3], we have

$$\begin{aligned}&\sup _{t\in [0,T]}|X^{\epsilon ,h_{\epsilon },n}(t)-X^{\epsilon ,h_{\epsilon }}(t)|^2\\&\quad \leqslant \sup _{t\in [0,T]}|M_{\epsilon }^n(t)\!-\!M_{\epsilon }(t)|^{1/2}\left( \sup _{t\in [0,T]}|M_{\epsilon }^n(t)\!-\!M_{\epsilon }(t)|\!+\!4|K^{\epsilon ,h_{\epsilon },n}|_T^0\!+\!4|K^{\epsilon ,h_{\epsilon }}|_T^0\right) ^{1/2}. \end{aligned}$$

By Lemma 4.2, we have

$$\begin{aligned} \mathbf{E}|K^{\epsilon ,h_{\epsilon }}|_T^0\leqslant C \end{aligned}$$

where $C$ is a constant which does not depend on $\epsilon$. We still need to verify

$$\begin{aligned} \mathbf{E}|K^{\epsilon ,h_{\epsilon },n}|_T^0\leqslant C, \end{aligned}$$

where $C$ is a constant which does not depend on $n$ and $\epsilon$. By Proposition 2.1 we have

$$\begin{aligned}&|X^{\epsilon ,h_{\epsilon },n}(t')-a|^2-|X^{\epsilon ,h_{\epsilon },n}(t)-a|^2\\&\quad \leqslant |M_{\epsilon }^n(t')+x-a|^2-|M_{\epsilon }^n(t)+x-a|^2\\&\qquad -\,\,2\gamma |K^{\epsilon ,h_{\epsilon },n}|_t^{t'}+2\gamma \mu (t'-t)+2\mu \int \limits _t^{t'}|X^{\epsilon ,h_{\epsilon },n}(s)-a|{\mathord {\mathrm{d}}}r\\&\qquad +\,\,2\int \limits _t^{t'}{\langle }M_{\epsilon }^n(s)-M_{\epsilon }(s),{\mathord {\mathrm{d}}}K^{\epsilon ,h_{\epsilon },n}(s){\rangle }_{{\mathbb {R}}^m}\\&\qquad +\,\,2{\langle }X^{\epsilon ,h_{\epsilon },n}(t')-x-M_{\epsilon }^n(t'),M_{\epsilon }^n(t')-M_{\epsilon }^n(t){\rangle }_{{\mathbb {R}}^m}\\&\quad \leqslant C\left( 1+\sup _{t\in [0,T]}|M_{\epsilon }(t)|^2+\sup _{t\in [0,T]}|X^{\epsilon ,h_{\epsilon },n}(t)-a|\right) \\&\qquad +\,\,2\left( \sup _{|s'-s|\leqslant t'-t}|M_{\epsilon }(s')-M_{\epsilon }(s)|-\gamma \right) |K^{\epsilon ,h_{\epsilon },n}|_{t'}^t. \end{aligned}$$

Hence

$$\begin{aligned}&|X^{\epsilon ,h_{\epsilon },n}(t')-a|^2-|X^{\epsilon ,h_{\epsilon },n}(t)-a|^2+\frac{\gamma }{2}|K^{\epsilon ,h_{\epsilon },n}|_{t'}^t\\&\quad \leqslant C\left( 1+\sup _{t\in [0,T]}|M_{\epsilon }(t)|^2+\sup _{t\in [0,T]}|X^{\epsilon ,h_{\epsilon },n}(t)-a|\right) \end{aligned}$$

on the set

$$\begin{aligned} {\varOmega }_{\epsilon ,n,\delta }=\left\{ \sup _{|s'-s|\leqslant \delta }|M_{\epsilon }^n (s')-M_{\epsilon }^n(s)|\leqslant \frac{\gamma }{2}\right\} . \end{aligned}$$

Consequently, applying the same procedure in [10, Proposition 4.9], we obtain

$$\begin{aligned} |K^{\epsilon ,h_{\epsilon },n}|_T^01_{{\varOmega }_{\epsilon ,n,\delta }} \leqslant \frac{C\Big (1+\sup _{t\in [0,T]}|M_{\epsilon }(t)|^2\Big )}{\delta }. \end{aligned}$$

For $p>6$, this yields

$$\begin{aligned} \mathbf{E}|K^{\epsilon ,h_{\epsilon },n}|_T^0&= \sum _{k=1}^{\infty }\mathbf{E}\left[ |K^{\epsilon ,h_{\epsilon },n}|_T^01_{{\varOmega }_{\epsilon ,n,\frac{1}{k+1}}\backslash {\varOmega }_{\epsilon ,n,\frac{1}{k}}}\right] \\&\leqslant C\sum _{k=1}^{\infty }(k+1)\mathbf{P}\left( {\varOmega }_{\epsilon ,n,\frac{1}{k}}^c\right) \\&\leqslant C\sum _{k=1}^{\infty }(k+1)\mathbf{E}\sup _{|t'-t|\leqslant \frac{1}{k}}|M_{\epsilon }(t')-M_{\epsilon }(t)|^p\\&\leqslant C\sum _{k=1}^{\infty }(k+1)\left( \frac{1}{k}\right) ^{p/2-1}<\infty . \end{aligned}$$

Therefore, we obtain the desired estimate, and hence (23) is satisfied.

Step 2

Similarly to Step 1, we can prove

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbf{E}\sup _{t\in [0,T]}|\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t)-\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha }(t)|=0 \end{aligned}$$

(24)

uniformly in $\epsilon$ and $\alpha$.

Step 3

Let $\alpha ,\beta >0$. Since $A_{\epsilon }^{\alpha }(x)\in A_{\epsilon }(J^{\alpha }(x))$, we have

$$\begin{aligned} {\langle }A_{\epsilon }^{\alpha }(x)-A_{\epsilon }^{\beta }(y),x-y{\rangle }_{{\mathbb {R}}^m}\geqslant -(\alpha +\beta ){\langle }A_{\epsilon }^{\alpha }(x),A_{\epsilon }^{\beta }(y){\rangle }_{{\mathbb {R}}^m},\quad \forall x,y\in {\mathbb {R}}^m. \end{aligned}$$

Therefore

$$\begin{aligned}&\sup _{t\in [0,T]}|\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t)-\widetilde{X}^{\epsilon ,h_{\epsilon },\beta ,n}(t)|^2\\&\quad =\sup _{t\in [0,T]}\left| \!-\!\int \limits _0^t{\langle }A_{\epsilon }^{\alpha }(\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(s))\!-\!A_{\epsilon }^{\beta }(\widetilde{X}^{\epsilon ,h_{\epsilon },\beta ,n}(s)),\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(s)\!-\!\widetilde{X}^{\epsilon ,h_{\epsilon },\beta ,n}(s){\rangle }_{{\mathbb {R}}^m}{\mathord {\mathrm{d}}}s\right| \\&\quad \leqslant (\alpha +\beta )\int \limits _0^T|A_{\epsilon }^{\alpha }(\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(s)||A_{\epsilon }^{\beta }(\widetilde{X}^{\epsilon ,h_{\epsilon },\beta ,n}(s)|{\mathord {\mathrm{d}}}s\\&\quad \leqslant (\alpha +\beta )T\sup _{t\in [0,T]}|A_{\epsilon }^{\alpha }(\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t)|\sup _{t\in [0,T]}|A_{\epsilon }^{\beta }(\widetilde{X}^{\epsilon ,h_{\epsilon },\beta ,n}(t)|. \end{aligned}$$

As in the proof of [10, Proposition 4.7], we have

$$\begin{aligned} \left| \frac{{\mathord {\mathrm{d}}}}{{\mathord {\mathrm{d}}}s}\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}\right| \leqslant |\dot{M}_{\epsilon }^n(t)|+|A_{\epsilon }^0(x)|+\int \limits _0^t|\ddot{M}_{\epsilon }^n(s)|{\mathord {\mathrm{d}}}s, \end{aligned}$$

and this gives

$$\begin{aligned} \sup _{t\in [0,T]}|A_{\epsilon }^{\alpha }(\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t))|\leqslant |A_{\epsilon }^0(x)|+2\sup _{t\in [0,T]}|\dot{M}_{\epsilon }^n(t)|+T\sup _{t\in [0,T]}|\ddot{M}_{\epsilon }^n(t)|. \end{aligned}$$

Consequently we have

$$\begin{aligned} \mathbf{E}\sup _{t\in [0,T]}|\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t)-\widetilde{X}^{\epsilon ,h_{\epsilon },\beta ,n}(t)|^2\leqslant C_n(\alpha +\beta ), \end{aligned}$$

where $C_n$ is a constant independent of $\epsilon$, $\alpha$ and $\beta$. We know from [10, Proposition 4.7] that

$$\begin{aligned} \sup _{t\in [0,T]}|\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t)-{X}^{\epsilon ,h_{\epsilon },n}(t)|\rightarrow 0\quad \text {a.s.}\quad \text {as}\quad \alpha \rightarrow 0. \end{aligned}$$

Therefore we have

$$\begin{aligned} \lim _{\alpha \rightarrow 0}\mathbf{E}\sup _{t\in [0,T]}|\widetilde{X}^{\epsilon ,h_{\epsilon },\alpha ,n}(t)-{X}^{\epsilon ,h_{\epsilon },n}(t)|^2=0 \end{aligned}$$

(25)

uniformly in $\epsilon$.

Then the proof is completed by (23)–(25). $\square$