Large deviations for backward stochastic differential equations driven by $G$-Brownian motion

The large deviation principle (LDP in short) characterizes the limiting behavior, as $\varepsilon\rightarrow 0$, of family of probability measures $\{\mu_{\varepsilon}\}_{\varepsilon>0}$ in terms of a rate function. Several authors have considered large deviations and obtained different types of applications mainly to mathematical physics. General references on large deviations are: Varadhan [1984], Deuschel and Stroock [1989], Dembo and Zeitouni [1998].

Let $X^{s,x,\varepsilon}$ be the diffusion process that is the unique solution of the following stochastic differential equation (SDE in short)

where $\beta$ is a Lipschitz function defined on $\mathbb{R}^{d}$ with values in $\mathbb{R}^{d}$, $\sigma$ is a Lipschitz function defined on $\mathbb{R}^{d}$ with values in $\mathbb{R}^{d\times k}$, and $W$ is a standard Brownian motion in $\mathbb{R}^{k}$ defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$. The existence and uniqueness of the strong solution $X^{s,x,\varepsilon}$ of (1.1) is standard. Thanks to the work of Freidlin and Wentzell [1984], the sequence $(X^{s,x,\varepsilon})_{\varepsilon>0}$ converges in probability, as $\varepsilon$ goes to 0, to $(\varphi_{t}^{s,x})_{s\leq t\leq T}$ solution of the following deterministic equation

and satisfies a large deviation principle (LDP in short).

Rainero [2006] extended this result to the case of backward stochastic differential equations (BSDEs in short) and Essaky [2008] and N’zi and Dakaou [2014] to reflected BSDEs.

Gao and Jiang [2010] extended the work of Freidlin and Wentzell [1984] to stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs in short). The authors considered the following $G$-SDE: for every $0\leq t\leq T$,

and use discrete time approximation to establish LDP for $G$-SDEs.

The aim of this paper is to establish LDP for $G$-BSDEs. More precisely, we consider the following forward-backward stochastic differential equation driven by $G$-Brownian motion: for every $s\leq t\leq T$,

We study the asymptotic behavior of the solution of the backward equation and establish a LDP for the corresponding process.

The remaining part of the paper is organized as follows. In Section 2, we present some preliminaries that are useful in this paper. Section 3 is devoted to the large deviations for stochastic differential equations driven by $G$-Brownian motion obtained by Gao and Jiang [2010]. The large deviations for backward stochastic differential equations driven by $G$-Brownian motion are given in Section 4.

We review some basic notions and results about $G$-expectation, $G$-Brownian motion and $G$-stochastic integrals [see Peng, 2010, Hu et al., 2014a; for more details].

Let $\Omega$ be a complete separable metric space, and let $\mathcal{H}$ be a linear space of real-valued functions defined on $\Omega$ satisfying: if $X_{i}\in\mathcal{H}$, $i=1,\ldots,n$, then

where $\mathcal{C}_{l,Lip}(\mathbb{R}^{n})$ is the space of real continuous functions defined on $\mathbb{R}^{n}$ such that for some $C>0$ and $k\in\mathbb{N}$ depending on $\varphi$,

Now we introduce the definition of $G$-normal distribution.

Throughout this paper, we consider only the non-degenerate case, i.e., $\underline{\sigma}^{2}>0$.

Let $\Omega:=\mathcal{C}\left([0,\infty)\right)$, which equipped with the raw filtration $\mathcal{F}$ generated by the canonical process $(B_{t})_{t\geq 0}$, i.e., $B_{t}(\omega)=\omega_{t}$, for $(t,\omega)\in[0,\infty)\times\Omega$. Let us consider the function spaces defined by

For $\xi\in Lip(\Omega_{T})$ and $p\geq 1$, we consider the norm $\|\xi\|_{L^{p}_{G}}:=\left(\widehat{\mathbb{E}}\Big{[}|\xi|^{p}\Big{]}\right)^{1/p}$. Denote by $L^{p}_{G}(\Omega_{T})$ the Banach completion of $Lip(\Omega_{T})$ under $\|\cdot\|_{L^{p}_{G}}$. It is easy to check that the conditional $G$-expectation $\widehat{\mathbb{E}}_{t}[\cdot]:Lip(\Omega_{T})\longrightarrow Lip(\Omega_{t})$ is a continuous mapping and thus can be extended to $\widehat{\mathbb{E}}_{t}[\cdot]:L^{p}_{G}(\Omega_{T})\longrightarrow L^{p}_{G}(\Omega_{t})$.

Let $\mathcal{P}$ be a weakly compact set that represents $\widehat{\mathbb{E}}$. For this $\mathcal{P}$, we define the capacity of a measurable set $A$ by

A set $A\in\mathcal{B}(\Omega_{T})$ is a polar if $\widehat{C}(A)=0$. A property holds quasi-surely (q.s.) if it is true outside a polar set.

An important feature of the $G$-expectation framework is that the quadratic variation $\left\langle B\right\rangle$ of the $G$-Brownian motion is no longer a deterministic process, which is given by

where $\pi_{t}^{N}=\{t_{0},t_{1},\ldots,t_{N}\}$, $N=1,2,\ldots$, are refining partitions of $[0,t]$. By Peng [2010], for all $s,t\geq 0$, $\langle B\rangle_{t+s}-\langle B\rangle_{t}\in[s\underline{\sigma}^{2},s\overline{\sigma}^{2}]$, $q.s.$

Let $M_{G}^{0}\left(0,T\right)$ be the collection of processes in the following form: for a given partition $\pi_{T}^{N}:=\{t_{0},t_{1},\ldots,t_{N}\}$ of $[0,T]$,

where $\xi_{i}\in Lip(\Omega_{t_{i}})$, for all $i=0,1,\ldots,N-1$. For $p\geq 1$ and $\eta\in M_{G}^{0}\left(0,T\right)$, let $\left\|\eta\right\|_{H_{G}^{p}}:=\left(\widehat{\mathbb{E}}\left[\left(\int_{0}^{T}|\eta_{s}|^{2}ds\right)^{p/2}\right]\right)^{1/p}$, $\|\eta\|_{M_{G}^{p}}:=\left(\widehat{\mathbb{E}}\left[\int_{0}^{T}|\eta_{s}|^{p}ds\right]\right)^{1/p}$ and denote by $H_{G}^{p}\left(0,T\right)$, $M_{G}^{p}(0,T)$ the completions of $M_{G}^{0}\left(0,T\right)$ under the norms $\|\cdot\|_{H_{G}^{p}}$, $\|\cdot\|_{M_{G}^{p}}$ respectively.

Let $\mathcal{S}_{G}^{0}\left(0,T\right):=\{h(t,B_{t_{1}\wedge t},B_{t_{2}\wedge t}-B_{t_{1}\wedge t},\ldots,B_{t_{n}\wedge t}-B_{t_{n-1}\wedge t}):0\leq t_{1}\leq t_{2}\leq\ldots\leq t_{n}\leq T,\leavevmode\nobreak\ h\in\mathcal{C}_{b,Lip}(\mathbb{R}^{n+1})\}$, where $\mathcal{C}_{b,Lip}(\mathbb{R}^{n+1})$ is the collection of all bounded and Lipschitz functions on $\mathbb{R}^{n+1}$. For $p\geq 1$ and $\eta\in\mathcal{S}_{G}^{0}\left(0,T\right)$, we set $\left\|\eta\right\|_{\mathcal{S}_{G}^{p}}:=\left(\widehat{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}|\eta_{t}|^{p}\Big{]}\right)^{1/p}$. We denote by $\mathcal{S}_{G}^{p}\left(0,T\right)$ the completion of $\mathcal{S}_{G}^{0}\left(0,T\right)$ under the norm $\left\|\cdot\right\|_{\mathcal{S}_{G}^{p}}$.

For $\xi\in Lip(\Omega_{T})$, let

$\mathcal{E}$ is called the $G$-evaluation.

For $p\geq 1$ and $\xi\in Lip(\Omega_{T})$, define

and denote by $L_{\mathcal{E}}^{p}(\Omega_{T})$ the completion of $Lip(\Omega_{T})$ under the norm $\|\cdot\|_{p,\mathcal{E}}$.

The following estimate will be used in this paper.

In this section, we present the large deviations for $G$-SDEs obtained by Gao and Jiang [2010]. The authors use discrete time approximation to obtain their results.

First, we recall the following notations on large deviations under a sublinear expectation.

Let $(\chi,d)$ be a Polish space. Let $\left(U^{\varepsilon},\;\varepsilon>0\right)$ be a family of measurable maps from $\Omega$ into $(\chi,d)$ and let $\delta(\varepsilon)$, $\varepsilon>0$ be a positive function satisfying $\delta(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$.

A nonnegative function $\mathcal{I}$ on $\chi$ is called to be (good) rate function if $\{x:\;\mathcal{I}(x)\leq\alpha\}$ (its level set) is (compact) closed for all $0\leq\alpha<\infty$.

$\left\{\widehat{C}(U^{\varepsilon}\in\cdot)\right\}_{\varepsilon>0}$ is said to satisfy large deviation principle with speed $\delta(\varepsilon)$ and with rate function $\mathcal{I}$ if for any measurable closed subset $\mathcal{F}\subset\chi$,

and for any measurable open subset $\mathcal{O}\subset\chi$,

In Gao and Jiang [2010], for any $\varepsilon>0$, the authors considered the following random perturbation SDEs driven by $d$-dimensional $G$-Brownian motion $B$

where $\langle B,B\rangle$ is treated as a $d\times d$-dimensional vector,

and $h^{\varepsilon}:\;\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n\times d^{2}}$.

Consider the following conditions:

(H1)

$b^{\varepsilon}$, $\sigma^{\varepsilon}$ and $h^{\varepsilon}$ are uniformly bounded;

(H2)

$b^{\varepsilon}$, $\sigma^{\varepsilon}$ and $h^{\varepsilon}$ are uniformly Lipschitz continuous;

(H3)

$b^{\varepsilon}$, $\sigma^{\varepsilon}$ and $h^{\varepsilon}$ converge uniformly to $b:=b^{0}$, $\sigma:=\sigma^{0}$ and $h:=h^{0}$ respectively.

Let $\mathcal{C}([0,T],\mathbb{R}^{n})$ be the space of $\mathbb{R}^{n}$-valued continuous functions $\varphi$ on $[0,T]$ and $\mathcal{C}_{0}([0,T],\mathbb{R}^{n})$ the space of $\mathbb{R}^{n}$-valued continuous functions $\widetilde{\varphi}$ on $[0,T]$ with $\widetilde{\varphi}_{0}=0$.

Define

We recall the following result of a joint large deviation principle for $G$-Brownian motion and its quadratic variation process.

For any $(\phi,\eta)\in\mathbb{H}^{d}\times\mathbb{A}$, let $\Psi(\phi,\eta)\in\mathcal{C}([0,T],\mathbb{R}^{n})$ be the unique solution of the following ordinary differential equation (ODE in short)

For $0\leq\alpha<1$ given and $n\geq 1$, for each $\psi\in\mathcal{C}_{0}([0,T],\mathbb{R}^{n})$, set

and

We immediately have the following result which will be used in the following section.