Moderate Deviations for the Langevin Equation with Strong Damping

Abstract

In this paper, we establish a moderate deviation principle for the Langevin dynamics with strong damping. The weak convergence approach plays an important role in the proof.

1 Introduction

For every $\varepsilon >0$, consider the following Langevin equation with strong damping

$$\begin{aligned} {\left\{ \begin{array}{ll} \ddot{q}^\varepsilon (t)=b(q^\varepsilon (t))-\frac{\alpha (q^\varepsilon (t))}{\varepsilon } {\dot{q}}^\varepsilon (t)+\sigma (q^\varepsilon (t)){\dot{B}}(t), \\ q^\varepsilon (0)=q \in \mathbb {R}^d,\quad {\dot{q}}^\varepsilon (0)=p \in \mathbb {R}^d. \end{array}\right. } \end{aligned}$$

(1.1)

Here B(t) is a d-dimensional standard Wiener process, defined on some complete stochastic basis $(\Omega ,{\mathcal {F}},\{{\mathcal {F}}_t\}_{{t \ge 0}},\mathbb {P})$. The coefficients $b, \alpha$ and $\sigma$ satisfy some regularity conditions (see Sect. 2 for details) such that for any fixed $\varepsilon>0,T>0$ and $k \ge 1$, Eq.(1.1) admits a unique solution $q^\varepsilon$ in $L^k(\Omega ;C([0,T];\mathbb {R}^d))$. Let $q_\varepsilon (t):= q^\varepsilon (t/\varepsilon )$, $t \ge 0$, then Eq. (1.1) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon ^2 \ddot{q}_\varepsilon (t)=b(q_\varepsilon (t))-\alpha (q_\varepsilon (t)) {\dot{q}}_\varepsilon (t) +\sqrt{\varepsilon }\sigma (q_\varepsilon (t)) {\dot{w}}(t), \\ q_\varepsilon (0)=q \in \mathbb {R}^d,\quad {\dot{q}}_\varepsilon (0)=\frac{p}{\varepsilon } \in \mathbb {R}^d, \end{array}\right. } \end{aligned}$$

(1.2)

where $w(t):=\sqrt{\varepsilon } B(t/\varepsilon )$, $t \ge 0$, is also a $\mathbb {R}^d$-valued Wiener process.

In [3], Cerrai and Freidlin established a large deviation principle (LDP for short) for Eq. (1.2) as $\varepsilon \rightarrow 0+$. More precisely, for any $T>0$, they proved that the family $\{q_\varepsilon \}_{\varepsilon >0}$ satisfies the LDP in the space $C([0,T]; \mathbb {R}^d)$, with the same rate function I and the same speed function $\varepsilon ^{-1}$ that describe the LDP of the first order equation

$$\begin{aligned} {\dot{g}}_\varepsilon (t)= \frac{b(g_\varepsilon (t))}{\alpha (g_\varepsilon (t))} +\sqrt{\varepsilon } \frac{\sigma (g_\varepsilon (t))}{ \alpha (g_\varepsilon (t))} {\dot{w}}(t),\ \ \ \ g_\varepsilon (0) = q \in \mathbb {R}^d. \end{aligned}$$

(1.3)

Explicitly, this means that

• (1) for any constant $c>0$, the level set $\{f; I(f)\le c\}$ is compact in $C([0,T];\mathbb {R}^d)$;

• (2) for any closed subset $F\subset C([0,T];\mathbb {R}^d)$,

  $$\begin{aligned} \limsup _{\varepsilon \rightarrow 0+}\varepsilon \log \mathbb {P}(q_{\varepsilon }\in F)\le -\inf _{f\in F}I(f); \end{aligned}$$

• (3) for any open subset $G\subset C([0,T];\mathbb {R}^d)$,

  $$\begin{aligned} \liminf _{\varepsilon \rightarrow 0+}\varepsilon \log \mathbb {P}(q_{\varepsilon }\in G)\ge -\inf _{f\in G}I(f). \end{aligned}$$

The dynamics system (1.3) can be regarded as the random perturbation of the following deterministic differential equation

$$\begin{aligned} {\dot{q}}_0(t) = \frac{b(q_0(t))}{\alpha (q_0(t))},\ \ \ q_0(0)=q \in \mathbb {R}^d. \end{aligned}$$

(1.4)

Roughly speaking, the LDP result in [3] shows that the asymptotic probability of $\mathbb {P}(\Vert q_{\varepsilon } -q_0\Vert \ge \delta )$ converges exponentially to 0 as $\varepsilon \rightarrow 0$ for any $\delta >0$, where $\Vert \cdot \Vert$ is the sup-norm on $C([0,T];\mathbb {R}^d)$.

Similarly to the large deviations, the moderate deviations arise in the theory of statistical inference quite naturally. The moderate deviation principle (MDP for short) can provide us with the rate of convergence and a useful method for constructing asymptotic confidence intervals (see, e.g., recent works [6, 8, 9, 11] and references therein). Usually, the quadratic form of the rate function corresponding to the MDP allows for the explicit minimization, and particularly it allows one to obtain an asymptotic evaluation for the exit time (see [10]). Recently, the study of the MDP estimates for stochastic (partial) differential equation has been carried out as well, see e.g. [1, 7, 12, 13] and so on.

In this paper, we shall investigate the MDP problem for the family $\{q_\varepsilon \}_{\varepsilon >0 }$ on $C([0,T];\mathbb {R}^d)$. That is, the asymptotic behavior of the trajectory

$$\begin{aligned} X_\varepsilon (t) = \frac{1}{\sqrt{\varepsilon } h(\varepsilon )} \left( q_\varepsilon (t)-q_{0}(t)\right) ,\quad t \in [0,T]. \end{aligned}$$

(1.5)

Here the deviation scale satisfies

$$\begin{aligned} h(\varepsilon )\rightarrow +\infty \ \text { and }\ \sqrt{\varepsilon } h(\varepsilon ) \rightarrow 0, \quad \text {as}\ \varepsilon \rightarrow 0. \end{aligned}$$

(1.6)

Due to the complexity of $q_\varepsilon$, we mainly use the weak convergence approach to deal with this problem. Comparing with the approximating method used in Gao and Wang [5], our method is simpler since we only need the moment estimation rather than the exponential moment estimation of the solution.

The organization of this paper is as follows. In Sect. 2, we present the framework of the Langevin equation, and then state our main results. Section 3 is devoted to proving the MDP.

2 Framework and Main Results

Let $|\cdot |$ be the Euclidean norm of a vector in $\mathbb {R}^d$, $\langle \cdot , \cdot \rangle$ the inner production in $\mathbb {R}^d$, and $\Vert \cdot \Vert _{\mathrm {HS}}$ the Hilbert-Schmidt norm in $\mathbb {R}^{d\times d}$ (the space of $d\times d$ matrices). For a function $b: \mathbb {R}^d\rightarrow \mathbb {R}^d$, $Db=\left( \frac{\partial }{\partial x_j} b^i\right) _{1\le i,j \le d}$ is the Jacobian matrix of b. Recall that $\Vert \cdot \Vert$ is the sup-norm on $C([0,T];\mathbb {R}^d)$. Throughout this paper, $T > 0$ is some fixed constant, $C(\cdot )$ is a positive constant depending on the parameters in the bracket and independent of $\varepsilon$. The value of $C(\cdot )$ may be different from line to line.

Assume that the coefficients $b,\alpha$ and $\sigma$ in (1.2) satisfy the following hypothesis.

Hypothesis 2.1

• (a) The mappings $b: \mathbb {R}^d \rightarrow \mathbb {R}^d$ and $\sigma : \mathbb {R}^d \rightarrow \mathbb {R}^{d\times d}$ are continuously differentiable, and there exists some constant $K>0$ such that for all $x, y\in \mathbb {R}^d$,

  $$\begin{aligned} |b(x)-b(y)| \le K|x-y|, \end{aligned}$$

  (2.1)

  and

  $$\begin{aligned} \Vert \sigma (x)-\sigma (y)\Vert _{\mathrm {HS}} \le K|x-y|,\ \Vert \sigma (x)\Vert _{\mathrm {HS}} \le K. \end{aligned}$$

  Moreover, the matrix $\sigma (q)$ is invertible for any $q \in \mathbb {R}^d$, and $\sigma ^{-1}: \mathbb {R}^d \rightarrow \mathbb {R}^{d\times d}$ is bounded.

• (b) The mapping $\alpha : \mathbb {R}^d \rightarrow \mathbb {R}$ belongs to $C_b^1(\mathbb {R}^d)$ and there exist some constants $0<\alpha _0\le \alpha _1$ and $K>0$ such that

  $$\begin{aligned} \alpha _0=\inf _{x \in \mathbb {R}^d} \alpha (x), \ \alpha _1=\sup _{x \in \mathbb {R}^d}\alpha (x) \text { and } \sup _{x \in \mathbb {R}^d}|\nabla \alpha (x)|\le K. \end{aligned}$$

Notice that:

• (1) $\Vert Db\Vert _{\mathrm {HS}}\le K$ since b is continuously differentiable and satisfies (2.1);

• (2) $\sigma /\alpha$ is Lipschitz continuous and bounded due to the Lipschitz-continuity and the boundness of the functions $\sigma$ and $1/\alpha$.

Under Hypothesis 2.1, according to [5], Theorem 2.2], we know that the family $\left\{ (g_\varepsilon -q_{0})/[\sqrt{\varepsilon }h(\varepsilon )]\right\} _{\varepsilon >0}$ satisfies the LDP on $C([0,T]; \mathbb {R}^d)$ with speed $h^2 (\varepsilon )$ and a good rate function I given by

$$\begin{aligned} I(\psi ) = \frac{1}{2} \inf _{h \in {\mathcal {H}};\psi =\Gamma _0(h)} \Vert h\Vert _{{\mathcal {H}}}^2, \end{aligned}$$

(2.2)

where

$$\begin{aligned} {\mathcal {H}}: =\left\{ h \in C([0,T];\mathbb {R}^d);\ h(t)=\int _0^t {\dot{h}}(s)ds,\ \Vert h\Vert _{\mathcal {H}}^2:= \int _0^T \left| {\dot{h}}(t)\right| ^2 dt <\infty \right\} \end{aligned}$$

(2.3)

and

$$\begin{aligned} \Gamma _0(h(t))= \int _0^t D\left( \frac{b(q_0(s))}{\alpha (q_0(s))}\right) \Gamma _0(h(s))ds +\int _0^t \frac{\sigma (q_0(s))}{\alpha (q_0(s))} {\dot{h}}(s) ds, \end{aligned}$$

(2.4)

with the convention $\inf \emptyset =\infty$. This special kind of LDP is just the MDP for the family $\{g_\varepsilon \}_{\varepsilon >0}$ (see [4]).

The main goal of this paper is to prove that the family $\{q_\varepsilon \}_{\varepsilon >0}$ satisfies the same MDP as the family $\{g_\varepsilon \}_{\varepsilon >0}$ on $C([0,T];\mathbb {R}^d)$.

Theorem 2.2

Under Hypothesis 2.1, the family $\{(q_\varepsilon -q_{0})/[\sqrt{\varepsilon }h(\varepsilon )] \}_{\varepsilon >0}$ obeys an LDP on $C([0,T]; \mathbb {R}^d)$ with the speed function $h^2(\varepsilon )$ and the rate function I given by (2.2).

3 Proof of MDP

3.1 Weak Convergence Approach in LDP

In this subsection, we will give the general criteria for the LDP given in [2].

Let $(\Omega ,{\mathcal {F}},\mathbb {P})$ be a probability space with an increasing family $\{{\mathcal {F}}_t\}_{0\le t\le T}$ of the sub-$\sigma$-fields of ${\mathcal {F}}$ satisfying the usual conditions. Let ${\mathcal {E}}$ be a Polish space with the Borel $\sigma$-field ${\mathcal {B}}({\mathcal {E}})$. The Cameron-Martin space associated with the Wiener process $\{w(t)\}_{0\le t\le T}$ (defined on the filtered probability space given above) is given by (2.3). See [4]. The space ${\mathcal {H}}$ is a Hilbert space with inner product

$$\begin{aligned} \langle h_1, h_2\rangle _{{\mathcal {H}}}:=\int _0^T\left\langle \dot{h}_1(s), \dot{h}_2(s)\right\rangle ds. \end{aligned}$$

Let ${\mathcal {A}}$ denote the class of all $\{{\mathcal {F}}_t\}_{0\le t \le T}$-predictable processes belonging to ${\mathcal {H}}$ a.s.. Define for any $N \in \mathbb {N}$,

$$\begin{aligned} S_N:=\left\{ h\in {\mathcal {H}}; \ \int _0^T \left| {\dot{h}}(s)\right| ^2ds\le N\right\} . \end{aligned}$$

Consider the weak convergence topology on ${\mathcal {H}}$, i.e., for any $h_n, h \in {\mathcal {H}}, n\ge 1$, $h_n$ converges weakly to h as $n \rightarrow +\infty$ if

$$\begin{aligned} \langle h_n-h,g\rangle _{\mathcal {H}}\rightarrow 0, \text{ as } n \rightarrow +\infty ,\ \forall g \in {\mathcal {H}}. \end{aligned}$$

It is easy to check that $S_N$ is a compact set in ${\mathcal {H}}$ under the weak convergence topology. Define

$$\begin{aligned} {\mathcal {A}}_N:=\left\{ \phi \in {\mathcal {A}};\ \phi (\omega )\in S_N,\ \mathbb {P}\text {-a.s.}\right\} . \end{aligned}$$

We present the following result from Budhiraja et al. [2].

Theorem 3.1

([2]) Let ${\mathcal {E}}$ be a Polish space with the Borel $\sigma$-field ${\mathcal {B}}({\mathcal {E}})$. For any $\varepsilon >0$, let $\Gamma _\varepsilon$ be a measurable mapping from $C([0,T];\mathbb {R}^d)$ into ${\mathcal {E}}$. Let $X_\varepsilon (\cdot ):=\Gamma _\varepsilon (w(\cdot ))$. Suppose there exists a measurable mapping $\Gamma _0:C([0,T];\mathbb {R}^d)\rightarrow {\mathcal {E}}$ such that

• (a) for every $N<+\infty$, the set

  $$\begin{aligned} \left\{ \Gamma _0\left( \int _0^{\cdot }\dot{h}(s)ds\right) ;\ h\in S_N\right\} \end{aligned}$$

  is a compact subset of ${\mathcal {E}}$;

• (b) for every $N<+\infty$ and any family $\{ h^\varepsilon \}_{\varepsilon >0}\subset {\mathcal {A}}_N$ satisfying that $h^\varepsilon$ (as $S_N$-valued random elements) converges in distribution to $h \in {\mathcal {A}}_N$ as $\varepsilon \rightarrow 0$,

  $$\begin{aligned} \Gamma _\varepsilon \left( w(\cdot )+\frac{1}{\sqrt{\varepsilon }}\int _0^{\cdot }\dot{h}^\varepsilon (s)ds\right) \ \text {converges to} \ \Gamma _0\left( \int _0^{\cdot }\dot{h}(s)ds\right) \end{aligned}$$

  in distribution as $\varepsilon \rightarrow 0$.

Then the family $\{X_\varepsilon \}_{\varepsilon >0}$ satisfies the LDP on ${\mathcal {E}}$ with the rate function I given by

$$\begin{aligned} I(g):=\inf _{h\in {\mathcal {H}};g=\Gamma _0\left( \int _0^{\cdot }\dot{h}(s)ds\right) }\left\{ \frac{1}{2}\int _0^T\left| \dot{h}(s)\right| ^2ds\right\} ,\ g\in {\mathcal {E}}, \end{aligned}$$

(3.1)

with the convention $\inf \emptyset =\infty$.

3.2 Reduction to the Bounded Case

Under Hypothesis 2.1, for every fixed $\varepsilon >0$, Eq. (1.2) admits a unique solution $q_\varepsilon$ in $L^k(\Omega ;C([0,T];\mathbb {R}^d))$. According to the proof of Theorem 3.3 in [3], we know that the solution $q_\varepsilon$ of Eq. (1.2) can be expressed in the following form:

$$\begin{aligned} q_\varepsilon (t)= q+ \int _0^t \frac{b(q_\varepsilon (s))}{\alpha (q_\varepsilon (s))} ds + \sqrt{\varepsilon } \int _0^t \frac{\sigma (q_\varepsilon (s))}{\alpha (q_\varepsilon (s))} d w(s)+R_\varepsilon (t), \end{aligned}$$

(3.2)

where

$$\begin{aligned} R_\varepsilon (t)&:=\frac{p}{\varepsilon }\int _0^t e^{-A_\varepsilon (s)}ds -\frac{1}{\alpha (q_\varepsilon (t))} \int _0^t e^{-A_\varepsilon (t,s)} b(q_\varepsilon (s))ds\nonumber \\&\quad +\int _0^t\left( \int _0^s e^{-A_\varepsilon (s,r)} b(q_\varepsilon (r)) dr \right) \frac{1}{\alpha ^2(q_\varepsilon (s))} \langle \nabla \alpha (q_\varepsilon (s)), {\dot{q}}_\varepsilon (s)\rangle ds\nonumber \\&\quad -\frac{1}{\alpha (q_\varepsilon (t))}H_\varepsilon (t) + \int _0^t \frac{1}{\alpha ^2(q_\varepsilon (s))} H_\varepsilon (s)\langle \nabla \alpha (q_\varepsilon (s)), {\dot{q}}_\varepsilon (s)\rangle ds\nonumber \\&{=:}\sum _{k=1}^5 I_\varepsilon ^k (t), \end{aligned}$$

(3.3)

with

$$\begin{aligned} \begin{aligned}&A_\varepsilon (t,s):=\frac{1}{\varepsilon ^2} \int _s^t \alpha (q_\varepsilon (r))dr,\quad A_\varepsilon (t):=A_\varepsilon (t,0),\\&H_\varepsilon (t): =\sqrt{\varepsilon } e^{-A_\varepsilon (t)} \int _0^t e^{A_\varepsilon (s)} \sigma (q_\varepsilon (s)) d w(s). \end{aligned} \end{aligned}$$

We denote the solution functional from $C([0,T];\mathbb {R}^d)$ into $C([0,T];\mathbb {R}^d)$ by ${\mathcal {G}}_{\varepsilon }$, i.e.,

$$\begin{aligned} {\mathcal {G}}_{\varepsilon }(w(t)):=q_\varepsilon (t),\ \forall t \in [0,T] . \end{aligned}$$

(3.4)

Let

$$\begin{aligned} X_{\varepsilon }(t):=\Gamma _\varepsilon (w(t)) :=\frac{{\mathcal {G}}_{\varepsilon }(w(t))-q_{0}(t)}{\sqrt{\varepsilon } h(\varepsilon )}, \ \forall t \in [0,T]. \end{aligned}$$

(3.5)

Then $X_\varepsilon$ solves the following equation

$$\begin{aligned} X_\varepsilon (t)&= \frac{1}{\sqrt{\varepsilon } h(\varepsilon )} \int _0^t \left[ \frac{b(q_{0}(s)+\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon (s))}{\alpha (q_{0}(s)+\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon (s))}-\frac{b(q_{0}(s))}{\alpha (q_{0}(s))}\right] ds\nonumber \\&\quad +\frac{1}{h(\varepsilon )} \int _0^t \frac{\sigma (q_{0}(s)+\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon (s))}{\alpha (q_{0}(s)+\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon (s))} dw(s)+\frac{R_\varepsilon (t)}{\sqrt{\varepsilon }h(\varepsilon )}, t\in [0,T]. \end{aligned}$$

(3.6)

We shall prove that $\{X_\varepsilon \}_{\varepsilon >0}$ obeys an LDP on $C([0,T]; \mathbb {R}^d)$ with speed function $h^2(\varepsilon )$ and the rate function I given by (2.2).

Since the family $\{q_\varepsilon \}_{\varepsilon >0}$ satisfies the LDP in the space $C([0,T]; \mathbb {R}^d)$ with the rate function I and the speed function $\varepsilon ^{-1}$ under Hypothesis 2.1 (see Cerrai and Freidlin [3]), there exist some positive constants R, C such that

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0} \varepsilon \log \mathbb {P}\left( \Vert q_\varepsilon \Vert \ge R \right) \le -C. \end{aligned}$$

Noticing (1.6), we have

$$\begin{aligned} \begin{aligned} \limsup _{\varepsilon \rightarrow 0} \frac{1}{h^2(\varepsilon )} \log \mathbb {P}\left( \Vert q_\varepsilon \Vert \ge R \right) =-\infty . \end{aligned} \end{aligned}$$

(3.7)

For any fixed constant $M>R$, define

$$\begin{aligned} b^M(x):={\left\{ \begin{array}{ll} b(x), \ &{}|x|< M;\\ g(x), \ &{}M\le |x|< M+1;\\ 0, \ &{}|x|\ge M+1, \end{array}\right. } \end{aligned}$$

where g(x) is some infinitely differentiable function on $\mathbb {R}^d$ such that $b^M(x)$ is continuous differentiable on $\mathbb {R}^d$. Then for all $t\in [0,T]$, we denote

$$\begin{aligned} \begin{aligned}&q_0^M(t):=q+\int _0^t \frac{b^M(q_0^M(s))}{\alpha (q_0^M(s))}ds;\\&q_\varepsilon ^M(t):= q+ \int _0^t \frac{b^M(q_\varepsilon ^M(s))}{\alpha (q_\varepsilon ^M(s))} ds + \sqrt{\varepsilon } \int _0^t \frac{\sigma (q_\varepsilon ^M(s))}{\alpha (q_\varepsilon ^M(s))} d w(s)+R^M_\varepsilon (t);\\&X_\varepsilon ^M(t):= \frac{1}{\sqrt{\varepsilon } h(\varepsilon )} \int _0^t \left[ \frac{b^M(q_{0}^M(s)+\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^M(s))}{\alpha (q_{0}^M(s)+\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^M(s))}-\frac{b^M(q_{0}^M(s))}{\alpha (q_{0}^M(s))}\right] ds\\&\quad \quad \quad \quad +\frac{1}{h(\varepsilon )} \int _0^t \frac{\sigma (q_{0}^M(s)+\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^M(s))}{\alpha (q_{0}^M(s)+\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^M(s))} dw(s)+\frac{R_\varepsilon ^M(t)}{\sqrt{\varepsilon }h(\varepsilon )}, \end{aligned} \end{aligned}$$

where the expression of $R^M_\varepsilon (t)$ is similar to Eq.  (3.3) with $b^M, q_\varepsilon ^M$ in place of $b, q_\varepsilon$.

Notice that $\Vert q_0\Vert$ is finite by the continuity of b and $\alpha$. Hence, we can choose M large enough such that $q_0(t)=q_0^M(t),\ \text{ for } \text{ all } \ t\in [0,T].$ Then for some M large enough, by Eq. (3.7), for all $\delta >0$, we have

$$\begin{aligned}&\limsup _{\varepsilon \rightarrow 0} \frac{1}{h^2(\varepsilon )}\log \mathbb {P}\left( \left\| X_\varepsilon -X_\varepsilon ^M\right\|>\delta \right) \nonumber \\ =&\limsup _{\varepsilon \rightarrow 0} \frac{1}{h^2(\varepsilon )}\log \mathbb {P}\left( \left\| \frac{q_\varepsilon -q_\varepsilon ^M}{\sqrt{\varepsilon }h(\varepsilon )}\right\|>\delta \right) \nonumber \\ \le&\limsup _{\varepsilon \rightarrow 0} \frac{1}{h^2(\varepsilon )}\log \mathbb {P}\left( \left\| q_\varepsilon -q_\varepsilon ^M\right\| >0\right) \nonumber \\ \le&\limsup _{\varepsilon \rightarrow 0} \frac{1}{h^2(\varepsilon )}\log \mathbb {P}(\Vert q_\varepsilon \Vert \ge M)=-\infty , \end{aligned}$$

(3.8)

which means that $X_\varepsilon$ is $h^2(\varepsilon )$-exponentially equivalent to $X_\varepsilon ^M$. Hence, to prove the LDP for $\{X_\varepsilon \}_{\varepsilon >0}$ on $C([0,T];\mathbb {R}^d)$, it is enough to prove that for $\{X_\varepsilon ^M\}_{\varepsilon >0}$, which is the task of the next part.

3.3 The LDP for $\{X_\varepsilon ^M\}_{\varepsilon >0}$

In this subsection, we prove that for some fixed constant M large enough , $\{X_\varepsilon ^M\}_{\varepsilon >0}$ obeys an LDP on $C([0,T]; \mathbb {R}^d)$ with speed function $h^2(\varepsilon )$ and the rate function I given by (2.2). Without loss of generality, we assume that b is bounded, i.e., $|b| \le K$ for some positive constant K. Then $\frac{b}{\alpha }$ is also Lipschitz continuous and bounded, and by the differentiability of $\frac{b}{\alpha }$, $D(\frac{b}{\alpha })$ is also bounded. From now on, we can drop the M in the notations for the sake of simplicity.

3.3.1 Skeleton Equations

For any $h\in {\mathcal {H}}$, consider the deterministic equation:

$$\begin{aligned} g^h(t)= \int _0^t D\left( \frac{b(q_0(s))}{\alpha (q_0(s))}\right) g^{h}(s)ds +\int _0^t \frac{\sigma (q_0(s))}{\alpha (q_0(s))} {\dot{h}}(s) ds. \end{aligned}$$

(3.9)

Lemma 3.2

Under Hypothesis 2.1, for any $h\in {\mathcal {H}}$, Eq. (3.9) admits a unique solution $g^h$ in $C([0,T];\mathbb {R}^d)$, denoted by $g^h(\cdot ){=:}\Gamma _0\left( \int _0^\cdot \dot{h}(s)ds\right)$. Moreover, for any $N>0$, there exists some positive constant $C(K,N,T,\alpha _0,\alpha _1)$ such that

$$\begin{aligned} \sup _{h\in S_N}\left\| g^h\right\| \le C(K,N,T,\alpha _0,\alpha _1). \end{aligned}$$

(3.10)

Proof

The existence and uniqueness of the solution can be proved similarly to the case of stochastic differential equation (1.3), but much more simply. (3.10) follows from the boundness conditions of the coefficient functions and Gronwall’s inequality. Here we omit the relative proof. $\square$

Proposition 3.3

Under Hypothesis 2.1, for every positive number $N<+\infty$, the family

$$\begin{aligned} K_N:= \left\{ \Gamma _0\left( \int _0^{\cdot }\dot{h}(s)ds\right) ; h\in S_N\right\} \end{aligned}$$

is compact in $C([0,T];\mathbb {R}^d)$.

Proof

To prove this proposition, it is sufficient to prove that the mapping $\Gamma _0$ defined in Lemma 3.2 is continuous from $S_N$ to $C([0,T];\mathbb {R}^d)$, since the fact that $K_N$ is compact follows from the compactness of $S_N$ under the weak topology and the continuity of the mapping $\Gamma _0$ from $S_N$ to $C([0,T];\mathbb {R}^d)$.

Assume that $h_n\rightarrow h$ weakly in $S_N$ as $n\rightarrow \infty$. We consider the following equation

$$\begin{aligned}&g^{h_n}(t)-g^h(t)\\ =&\int _0^tD\left( \frac{b(q_0(s))}{\alpha (q_0(s))}\right) \left( g^{h_n}(s)-g^h(s)\right) ds+\int _0^t \frac{\sigma (q_0(s))}{\alpha (q_0(s))}\left( \dot{h}_n(s)-\dot{h}(s)\right) ds\\ {=:}&I^n_1(t)+I^n_2(t). \end{aligned}$$

Due to Cauchy-Schwartz inequality and the boundness of functions $\sigma ,\alpha$, we know that for any $0 \le t_1 \le t_2 \le T$,

$$\begin{aligned} |I_2^n(t_2)-I_2^n(t_1)| =&\left| \int _{t_1}^{t_2}\frac{\sigma (q_0(s))}{\alpha (q_0(s))} \left( \dot{h}_n(s)-\dot{h}(s)\right) ds\right| \nonumber \\ \le&\left( \int _{t_1}^{t_2} \left\| \frac{\sigma (q_0(s))}{\alpha (q_0(s))}\right\| ^2_{HS}ds\right) ^{\frac{1}{2}}\cdot \left( \int _{t_1}^{t_2} \left| \dot{h}_n(s)-\dot{h}(s)\right| ^2ds\right) ^{\frac{1}{2}}\nonumber \\ \le&C(K, \alpha _0)N^{\frac{1}{2}}(t_2-t_1)^{\frac{1}{2}}. \end{aligned}$$

(3.11)

Hence, the family of functions $\{I_2^n\}_{n\ge 1}$ is equiv-continuous in $C([0,T];{\mathbb {R}}^d)$. Particularly, taking $t_1=0$, we obtain that

$$\begin{aligned} \left\| I^n_2\right\| \le&C( K, N, T,\alpha _0)<\infty , \end{aligned}$$

(3.12)

where $C( K, N, T,\alpha _0)$ is independent of n. Thus, by the Ascoli-Arzelá theorem, the set $\{I_2^n\}_{n\ge 1}$ is compact in $C([0,T];\mathbb {R}^d)$.

On the other hand, for any $v\in \mathbb {R}^d$, by the boundness of $\sigma /\alpha$, we know that the function $\frac{\sigma (q_0)}{\alpha (q_0)}v$ belongs to $L^2([0,T];\mathbb {R}^d)$. Since $\dot{h}_n\rightarrow \dot{h}$ weakly in $L^2([0,T];\mathbb {R}^d)$ as $n \rightarrow +\infty$, we know that

$$\begin{aligned} \left\langle I^n_2(t),v\right\rangle =\int _0^t \frac{\sigma (q_0(s))}{\alpha (q_0(s))}\left( \dot{h}_n(s)-\dot{h}(s)\right) vds\rightarrow 0, \ \ \text {as } n\rightarrow \infty . \end{aligned}$$

(3.13)

Then by the compactness of $\{I_2^n\}_{n\ge 1}$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty } \left\| I^n_2 \right\| =0. \end{aligned}$$

(3.14)

Set $\zeta ^n(t)=\sup _{0\le s\le t}\left| g^{h_n}(s)-g^h(s)\right|$. By the boundness of $D(b/\alpha )$, we have

$$\begin{aligned} \zeta ^n(t)\le C(K,\alpha _0,\alpha _1)\int _0^t \zeta ^n(s) ds+\left\| I^n_2\right\| . \end{aligned}$$

By Gronwall’s inequality and (3.14), we have

$$\begin{aligned} \left\| g ^{h_n}-g^h\right\| \le e^{C(K,\alpha _0,\alpha _1)T}\cdot \left\| I^n_2\right\| \rightarrow 0, \text { as } n\rightarrow \infty , \end{aligned}$$

which completes the proof. $\square$

3.3.2 MDP

For any predictable process ${\dot{u}}$ taking values in $L^2 ([0,T]; \mathbb {R}^d)$, we denote by $q_\varepsilon ^u(t)$ the solution of the following equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon ^2 \ddot{q}_\varepsilon ^u (t)=b(q_\varepsilon ^u(t))-\alpha (q_\varepsilon ^u(t)) {\dot{q}}_\varepsilon ^u (t) +\sqrt{\varepsilon }\sigma (q_\varepsilon ^u(t)) {\dot{w}}(t)\\ +\sqrt{\varepsilon } h(\varepsilon )\sigma (q_\varepsilon ^u(t) ){\dot{u}}(t), \ t \in [0,T],\\ q_\varepsilon ^u(0)=q \in \mathbb {R}^d,\quad {\dot{q}}_\varepsilon ^u(0)=\frac{p}{\varepsilon } \in \mathbb {R}^d. \end{array}\right. } \end{aligned}$$

(3.15)

As is well known, for any fixed $\varepsilon >0$, $T>0$ and $k\ge 1$, this equation admits a unique solution $q_\varepsilon ^u$ in $L^k(\Omega ; C([0,T];\mathbb {R}^d))$ as follows

$$\begin{aligned} q_\varepsilon ^{u}(t)={\mathcal {G}}_\varepsilon \left( w(t)+h(\varepsilon )\int _0^{t}\dot{u}(s)ds\right) , \end{aligned}$$

where ${\mathcal {G}}_\varepsilon$ is defined by (3.4).

Lemma 3.4

Under Hypothesis 2.1, for every fixed $N\in \mathbb {N}$ and $\varepsilon >0$, let $u^\varepsilon \in {\mathcal {A}}_N$ and $\Gamma _\varepsilon$ be given by (3.5). Then $X_\varepsilon ^{u^\varepsilon }(\cdot ):=\Gamma _\varepsilon \left( w(\cdot )+h(\varepsilon )\int _0^{\cdot }\dot{u}^\varepsilon (s)ds\right)$ is the unique solution of the following equation

$$\begin{aligned} X_\varepsilon ^{u^\varepsilon } (t)&= \int _0^t \frac{1}{\sqrt{\varepsilon } h(\varepsilon )} \left[ \frac{b( q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s) )}{ \alpha (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} - \frac{b(q_{0} (s))}{\alpha (q_{0}(s))} \right] ds\nonumber \\&\quad + \int _0^t \frac{\sigma (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}{\alpha (q_{0}(s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} {\dot{u}}^\varepsilon (s) ds\nonumber \\&\quad + \frac{1}{h(\varepsilon )} \int _0^t \frac{\sigma (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}{ \alpha (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} dw(s)+\frac{R_\varepsilon ^{u^\varepsilon }(t)}{\sqrt{\varepsilon } h(\varepsilon )}, \quad t\in [0,T], \end{aligned}$$

(3.16)

where

$$\begin{aligned} \begin{aligned} R_\varepsilon ^{u^\varepsilon } (t)&=\frac{p}{\varepsilon }\int _0^t e^{-A_\varepsilon ^{u^\varepsilon } (s)}ds -\frac{1}{\alpha (q_\varepsilon ^{u^\varepsilon }(t))} \int _0^t e^{-A_\varepsilon ^{u^\varepsilon }(t,s)} b(q_\varepsilon ^{u^\varepsilon }(s))ds\\&\quad +\int _0^t\left( \int _0^s e^{-A_\varepsilon ^{u^\varepsilon }(s,r)} b(q_\varepsilon ^{u^\varepsilon }(r)) dr \right) \frac{1}{\alpha ^2(q_\varepsilon ^{u^\varepsilon }(s))} \left\langle \nabla \alpha (q_\varepsilon ^{u^\varepsilon }(s)), {\dot{q}}_\varepsilon ^{u^\varepsilon } (s)\right\rangle ds\\&\quad -\frac{1}{\alpha (q_\varepsilon ^{u^\varepsilon }(t))}H_\varepsilon ^{1,u^\varepsilon } (t) + \int _0^t \frac{1}{\alpha ^2(q_\varepsilon ^{u^\varepsilon }(s))} H_\varepsilon ^{1,u^\varepsilon }(s)\left\langle \nabla \alpha (q_\varepsilon ^{u^\varepsilon }(s)), {\dot{q}}_\varepsilon ^{u^\varepsilon } (s)\right\rangle ds\\&\quad -\frac{1}{\alpha (q_\varepsilon ^{u^\varepsilon }(t))}H_\varepsilon ^{2,u^\varepsilon } (t) + \int _0^t \frac{1}{\alpha ^2(q_\varepsilon ^{u^\varepsilon }(s))} H_\varepsilon ^{2,u^\varepsilon }(s)\left\langle \nabla \alpha (q_\varepsilon ^{u^\varepsilon }(s)), {\dot{q}}_\varepsilon ^{u^\varepsilon } (s)\right\rangle ds\\&{=:}\sum _{k=1}^7 I_\varepsilon ^{k,u^\varepsilon }, \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned}&A_\varepsilon ^{u^\varepsilon } (t,s):=\frac{1}{\varepsilon ^2} \int _s^t \alpha (q_\varepsilon ^{u^\varepsilon }(r))dr,\quad A_\varepsilon ^{u^\varepsilon }(t)=A_\varepsilon ^{u^\varepsilon }(t,0),\\&H_\varepsilon ^{1,u^\varepsilon }(t):=\sqrt{\varepsilon } e^{-A_\varepsilon ^{u^\varepsilon }(t)} \int _0^t e^{A_\varepsilon ^{u^\varepsilon } (s)} \sigma (q_\varepsilon ^{u^\varepsilon }(s)) d w(s),\\&H_\varepsilon ^{2,u^\varepsilon }(t):=\sqrt{\varepsilon }h(\varepsilon ) e^{-A_\varepsilon ^{u^\varepsilon }(t)} \int _0^t e^{A_\varepsilon ^{u^\varepsilon } (s)} \sigma (q_\varepsilon ^{u^\varepsilon }(s)) {\dot{u}}^\varepsilon (s)ds. \end{aligned} \end{aligned}$$

(3.17)

Furthermore, there exists a positive constant $\varepsilon _0 >0$ such that for any $\varepsilon \in (0,\varepsilon _0]$,

$$\begin{aligned} \mathbb {E}\left[ \int _0^T \left| X_\varepsilon ^{u^\varepsilon } (t)\right| ^2dt\right] \le C(K,N,T,\alpha _0,\alpha _1,|p|,|q|). \end{aligned}$$

(3.18)

Moveover, we have

$$\begin{aligned} \mathbb {E}\left[ \left\| X_\varepsilon ^{u^\varepsilon }\right\| ^2\right] \le C(K,N,T,\alpha _0,\alpha _1,|p|,|q|). \end{aligned}$$

(3.19)

To prove Lemma 3.4 and our main result, we present the following three lemmas. The first lemma is similar to [3, Lemma 3.1].

Lemma 3.5

Under Hypothesis 2.1, for any $T>0$, $k\ge 1$ and $N>0$, there exists some constant $\varepsilon _0>0$ such that for any $u^\varepsilon \in {\mathcal {A}}_N$ and $\varepsilon \in (0,\varepsilon _0]$, we have

$$\begin{aligned} \sup _{t \in [0,T] } \mathbb {E}\left[ \left| H_{\varepsilon }^{1,u^{\varepsilon }}(t)\right| ^k\right]\le & {} C(k,K ,N, T,\alpha _0,\alpha _1)\left( |q|^k+|p|^k +1\right) \varepsilon ^{\frac{3k}{2}}\nonumber \\&\quad +C(k,K)\varepsilon ^{\frac{k}{2}} t^{\frac{k}{2}} e^{-\frac{k\alpha _0 t}{\varepsilon ^2}}. \end{aligned}$$

(3.20)

Moveover, we have

$$\begin{aligned} \mathbb {E}\left\| H_\varepsilon ^{1,u^\varepsilon }\right\| \le \sqrt{\varepsilon } C(K,N,T,\alpha _0,\alpha _1)(1+|q|+|p|). \end{aligned}$$

(3.21)

Proof

Notice that Eq. (3.15) can be rewritten as the following equation: for all $t \in [0,T]$,

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{q}}_\varepsilon ^{u^\varepsilon }(t)=p_\varepsilon ^{u^\varepsilon }(t),\\ \varepsilon ^2{\dot{p}}_\varepsilon ^{u^\varepsilon }(t)=b(q_\varepsilon ^{u^\varepsilon }(t))-\alpha (q_\varepsilon ^{u^\varepsilon }(t))p_\varepsilon ^{u^\varepsilon }(t)+\sqrt{\varepsilon }\sigma (q_\varepsilon ^{u^\varepsilon }(t)){\dot{w}}(t)+ \sqrt{\varepsilon }h(\varepsilon )\sigma (q_\varepsilon ^{u^\varepsilon }(t)){\dot{u}}^\varepsilon (t),\\ q_\varepsilon ^{u^\varepsilon }(0)=q\in \mathbb {R}^d, \ p_\varepsilon ^{u^\varepsilon }(0)=\frac{p}{\varepsilon } \in \mathbb {R}^d. \end{array}\right. } \end{aligned}$$

From the notation given in Eq. (3.17), we have

$$\begin{aligned} {\dot{q}}_\varepsilon ^{u^\varepsilon }(t)= & {} p_\varepsilon ^{u^\varepsilon }(t) =\frac{1}{\varepsilon }e^{-A_\varepsilon ^{u^\varepsilon }(t)} p+\frac{1}{\varepsilon ^2}\int _0^t e^{-A_\varepsilon ^{u^\varepsilon }(t,s)}b(q_\varepsilon ^{u^\varepsilon }(s))ds+ \frac{1}{\varepsilon ^2}H_\varepsilon ^{2,u^\varepsilon }(t)\nonumber \\&\quad +\frac{1}{\varepsilon ^2}H_\varepsilon ^{1,u^\varepsilon }(t). \end{aligned}$$

(3.22)

Integrating with respect to t, we obtain that

$$\begin{aligned} \begin{aligned} q_\varepsilon ^{u^\varepsilon }(t)&=q+\frac{1}{\varepsilon }\int _0^te^{-A_\varepsilon ^{u^\varepsilon }(s)} pds+\frac{1}{\varepsilon ^2}\int _0^t\int _0^s e^{-A_\varepsilon ^{u^\varepsilon }(s,r)}b(q_\varepsilon ^{u^\varepsilon }(r))drds\\&\quad +\frac{1}{\varepsilon ^2}\int _0^t H_\varepsilon ^{2,u^\varepsilon }(s)ds+\frac{1}{\varepsilon ^2}\int _0^t H_\varepsilon ^{1,u^\varepsilon }(s)ds. \end{aligned} \end{aligned}$$

By Hypothesis 2.1 and Young’s inequality for integral operators, we have

$$\begin{aligned} \begin{aligned} \left| q_\varepsilon ^{u^\varepsilon }(t)\right|&\le |q|+\frac{\varepsilon }{\alpha _0} |p|+C(K,T,\alpha _0)\int _0^t(1+\left| q_\varepsilon ^{u^\varepsilon }(s)\right| )ds\\&\quad +C(K,\alpha _0)\sqrt{\varepsilon }h(\varepsilon )\int _0^t\left| {\dot{u}}^\varepsilon (s)\right| ds+\frac{1}{\varepsilon ^2}\int _0^t \left| H_\varepsilon ^{1,u^\varepsilon }(s)\right| ds\\&\le C(K,N,T,\alpha _0)\left( |q|+\varepsilon |p|+\sqrt{\varepsilon }h(\varepsilon ) \right) \nonumber \\&\quad +\frac{1}{\varepsilon ^2}\int _0^t \left| H_\varepsilon ^{1,u^\varepsilon }(s)\right| ds+C(K,T,\alpha _0)\int _0^t\left| q_\varepsilon ^{u^\varepsilon }(s)\right| ds. \end{aligned} \end{aligned}$$

Since $\lim _{\varepsilon \rightarrow 0}\sqrt{\varepsilon }h(\varepsilon )=0$, for $\varepsilon$ small enough, by Gronwall’s inequality,

$$\begin{aligned} \left| q_\varepsilon ^{u^\varepsilon }(t)\right| \le C(K,N,T,\alpha _0)(\left| q\right| +\left| p\right| +1) +C(K,T,\alpha _0)\frac{1}{\varepsilon ^2}\int _0^t \left| H_\varepsilon ^{1,u^\varepsilon }(s)\right| ds.\qquad \end{aligned}$$

(3.23)

Hence by the similar proof to that in [3, Lemma 3.1], we obtain (3.20) and (3.21). $\square$

For $H_\varepsilon ^{2,u^\varepsilon }(t)$, we have the following estimation.

Lemma 3.6

Under Hypothesis 2.1, for any $T>0$, $k\ge 1$ and $N\in \mathbb {N}$, there exists some constant $\varepsilon _0>0$ such that for any $u^\varepsilon \in {\mathcal {A}}_N$ and $\varepsilon \in (0,\varepsilon _0]$, we have

$$\begin{aligned} \mathbb {E}\left[ \left\| H_\varepsilon ^{2,u^\varepsilon }\right\| ^k\right] \le C(K,N,\alpha _0)\varepsilon ^{\frac{3k}{2}} h^k(\varepsilon ). \end{aligned}$$

(3.24)

Proof

For any $t\in [0,T]$ and $u^\varepsilon \in {\mathcal {A}}_N$, by the boundness of $\sigma$ and Cauchy-Schwarz inequality, we have

$$\begin{aligned} \begin{aligned} \left| H_\varepsilon ^{2,u^\varepsilon }(t)\right|&=\left| \sqrt{\varepsilon }h(\varepsilon ) e^{-A_\varepsilon ^{u^\varepsilon }(t)} \int _0^t e^{A_\varepsilon ^{u^\varepsilon } (s)} \sigma (q_\varepsilon ^{u^\varepsilon }(s)) {\dot{u}}^\varepsilon (s)ds \right| \\&\le K \sqrt{\varepsilon }h(\varepsilon )e^{-A_\varepsilon ^{u^\varepsilon }(t)} \int _0^t e^{A_\varepsilon ^{u^\varepsilon } (s)} \left| {\dot{u}}^\varepsilon (s)\right| ds\\&\le K \sqrt{\varepsilon }h(\varepsilon )e^{-A_\varepsilon ^{u^\varepsilon }(t)} \left( \int _0^t e^{2A_\varepsilon ^{u^\varepsilon } (s)} ds\right) ^{\frac{1}{2}} \left( \int _0^T \left| {\dot{u}}^\varepsilon (s)\right| ^2ds \right) ^{\frac{1}{2}}\\&\le KN^{\frac{1}{2}}\sqrt{\varepsilon }h(\varepsilon )e^{-A_\varepsilon ^{u^\varepsilon }(t)} \left( \int _0^t e^{2A_\varepsilon ^{u^\varepsilon } (s)} ds\right) ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

Since $A_\varepsilon ^{u^\varepsilon } (t)=\frac{1}{\varepsilon ^2} \int _0^t \alpha (q_\varepsilon ^{u^\varepsilon }(r))dr$, we have

$$\begin{aligned} \begin{aligned} \int _0^t e^{2A_\varepsilon ^{u^\varepsilon } (s)} ds&=\int _0^t \frac{\varepsilon ^2}{ 2\alpha (q_\varepsilon ^{u^\varepsilon }(s))} de^{ \frac{2}{\varepsilon ^2} \int _0^s \alpha (q_\varepsilon ^{u^\varepsilon } (r))dr}\\&\le \frac{\varepsilon ^2}{ 2 \alpha _0} \int _0^t de^{ \frac{2}{\varepsilon ^2} \int _0^s \alpha (q_\varepsilon ^{u^\varepsilon } (r))dr}\\&=\frac{\varepsilon ^2}{ 2 \alpha _0}\left( e^{2A_\varepsilon ^{u^\varepsilon }(t)}-1\right) . \end{aligned} \end{aligned}$$

Hence

$$\begin{aligned} \begin{aligned} \left| H_\varepsilon ^{2,u^\varepsilon }(t)\right|&\le KN^{\frac{1}{2}}\frac{\varepsilon ^\frac{3}{2} h(\varepsilon )}{ \sqrt{2 \alpha _0}}e^{-A_\varepsilon ^{u^\varepsilon }(t)}\left( e^{2A_\varepsilon ^{u^\varepsilon }(t)}-1\right) ^\frac{1}{2}\\&\le C(K,N,\alpha _0) \varepsilon ^{\frac{3}{2}} h(\varepsilon ) e^{-A_\varepsilon ^{u^\varepsilon }(t)}e^{A_\varepsilon ^{u^\varepsilon }(t)}\\&=C(K,N,\alpha _0)\varepsilon ^{\frac{3}{2}} h(\varepsilon ), \end{aligned} \end{aligned}$$

and furthermore

$$\begin{aligned} \mathbb {E}\left\| H_\varepsilon ^{2,u^\varepsilon } \right\| ^k\le C(K,N,\alpha _0)\varepsilon ^{\frac{3k}{2}} h^k(\varepsilon ), \end{aligned}$$

which completes the proof. $\square$

Lemma 3.7

Under Hypothesis 2.1, for any $T>0$ and any $u^\varepsilon \in {\mathcal {A}}_N$, we have

$$\begin{aligned} \mathbb {E}\left\| \frac{R_\varepsilon }{\sqrt{\varepsilon }h(\varepsilon )}\right\| \rightarrow 0, \ \ \ \text { as } \ \ \varepsilon \rightarrow 0. \end{aligned}$$

(3.25)

Moreover, we have

$$\begin{aligned} \mathbb {E}\left[ \left\| \frac{R_\varepsilon }{\sqrt{\varepsilon }h(\varepsilon )}\right\| ^2\right] \rightarrow 0, \ \ \ \text { as } \ \ \varepsilon \rightarrow 0. \end{aligned}$$

(3.26)

Proof

Similarly to the proof [3, (3.17)], under Hypothesis 2.1, we have

$$\begin{aligned} \mathbb {E}\left\| \frac{\sum _{k=1}^5 I_\varepsilon ^{k,u^\varepsilon }}{\sqrt{\varepsilon }h(\varepsilon )}\right\| \le \frac{1}{h(\varepsilon )}C(K,N,T,\alpha _0,\alpha _1,|p|,|q|) \rightarrow 0, \ \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

(3.27)

Next, we will estimate $\mathbb {E}\left\| \frac{I_\varepsilon ^{6,u^\varepsilon }}{\sqrt{\varepsilon } h(\varepsilon )}\right\|$ and $\mathbb {E}\left\| \frac{I_\varepsilon ^{7,u^\varepsilon }}{\sqrt{\varepsilon } h(\varepsilon )}\right\|$. By Lemma 3.6, we have

$$\begin{aligned} \mathbb {E}\left\| \frac{I_\varepsilon ^{6,u^\varepsilon }}{\sqrt{\varepsilon } h(\varepsilon )}\right\| \le \frac{1}{\sqrt{\varepsilon } h(\varepsilon )\alpha _0} \mathbb {E}\left\| H_\varepsilon ^{2,u^\varepsilon }\right\| \le \varepsilon C(K,N,\alpha _0) \rightarrow 0, \ \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

(3.28)

By Cauchy-Schwarz inequality, we have

$$\begin{aligned} \begin{aligned} \mathbb {E}\left\| \frac{I_\varepsilon ^{7,u^\varepsilon }}{\sqrt{\varepsilon } h(\varepsilon )}\right\|&\le \frac{C(K,\alpha _0)}{ \sqrt{\varepsilon } h(\varepsilon )} \mathbb {E}\left[ \sup _{t\in [0,T]} \int _0^t \left| H_\varepsilon ^{2,u^\varepsilon }(s)\right| \cdot \left| {\dot{q}}_\varepsilon ^{u^\varepsilon }(s)\right| ds \right] \\&\le \frac{C(K,\alpha _0)}{ \sqrt{\varepsilon } h(\varepsilon )} \left[ \int _0^T \mathbb {E}\left[ \left| H_\varepsilon ^{2,u^\varepsilon }(s)\right| ^2 \right] ds \right] ^{\frac{1}{2}}\cdot \left[ \int _0^T \mathbb {E}\left[ \left| {\dot{q}}_\varepsilon ^{u^\varepsilon }(s)\right| ^2\right] ds \right] ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

By (3.23), we have for all $\varepsilon >0$ small enough,

$$\begin{aligned} \int _0^T \left| {\dot{q}}_\varepsilon ^{u^\varepsilon }(s)\right| ^2 ds \le C(K,N,T,\alpha _0,|p|,|q|)+\frac{C(K,T,\alpha _0)}{\varepsilon ^4}\int _0^T \left| H_\varepsilon ^{1,u^\varepsilon }(s)\right| ^2 ds. \end{aligned}$$

Hence, by (3.20) and Lemma 3.6, we have

$$\begin{aligned}&\quad \mathbb {E}\left\| \frac{I_\varepsilon ^{7,u^\varepsilon }}{\sqrt{\varepsilon } h(\varepsilon )}\right\| \nonumber \\&\le \frac{C(K,N,T,\alpha _0,|p|,|q|)}{\sqrt{\varepsilon } h(\varepsilon )}\left[ \left( \int _0^T \mathbb {E}\left[ \left| H_\varepsilon ^{2,u^\varepsilon }(s)\right| ^2\right] ds\right) ^{\frac{1}{2}}\right] \nonumber \\&\quad + \frac{C(K,N,T,\alpha _0)}{\varepsilon ^{\frac{5}{2}} h(\varepsilon )}\left( \int _0^T \mathbb {E}\left[ \left| H_\varepsilon ^{2,u^\varepsilon }(s)\right| ^2\right] ds\right) ^{\frac{1}{2}}\cdot \left( \int _0^T \mathbb {E}\left[ \left| H_\varepsilon ^{1,u^\varepsilon }(s)\right| ^2\right] ds\right) ^{\frac{1}{2}} \nonumber \\&\le \sqrt{\varepsilon } C(K,N,T,\alpha _0,\alpha _1,|p|,|q|) \rightarrow 0,\ \ \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

(3.29)

This together with (3.27) and (3.28) implies (3.25).

(3.26) can be easily obtained by applying the similar estimation process for

$$\begin{aligned} \mathbb {E}\left[ \left\| \frac{I_\varepsilon ^{i,u^\varepsilon }}{\sqrt{\varepsilon } h(\varepsilon )}\right\| ^2\right] ,\ i=1,2,3,\cdots ,7, \end{aligned}$$

as given above. Hence we omit the proof. $\square$

Now we prove Lemma 3.4.

The proof of Lemma 3.4

For any $\varepsilon >0$ and $u^\varepsilon \in {\mathcal {A}}_N$, define

$$\begin{aligned} d\mathbb {Q}^{u^\varepsilon } := \exp \left\{ -h(\varepsilon ) \int _0^t {\dot{u}}^\varepsilon (s)dw(s)-\frac{ h^2(\varepsilon )}{2} \int _0^t \left| \dot{u}^{\varepsilon }(s)\right| ^2 ds\right\} d\mathbb {P}. \end{aligned}$$

Since $\frac{d\mathbb {Q}^{u^{\varepsilon }}}{d\mathbb {P}}$ is an exponential martingale, $\mathbb {Q}^{u^\varepsilon }$ is a probability measure on $\Omega$. By Girsanov theorem, the process

$$\begin{aligned} \tilde{w}^{\varepsilon }(t)=w(t)+h(\varepsilon )\int _0^t \dot{u}^{\varepsilon }(s)ds \end{aligned}$$

is a $\mathbb {R}^d$-valued Wiener process under the probability measure $\mathbb {Q}^{u^\varepsilon }$. Rewriting Eq. (3.16) with $\tilde{w}^{\varepsilon }(t)$, we obtain Eq. (3.6) with $\tilde{w}^{\varepsilon }(t)$ in place of w(t). Let $X_\varepsilon ^{u^\varepsilon }$ be the unique solution of Eq. (3.6) with $\tilde{w}^{\varepsilon }(t)$ on the space $(\Omega ,{\mathcal {F}},\mathbb {Q}^{u^\varepsilon })$. Then $X_\varepsilon ^{u^\varepsilon }$ satisfies Eq. (3.16), $\mathbb {Q}^{u^\varepsilon }$-a.s.. By the equivalence of probability measures, $X_\varepsilon ^{u^\varepsilon }$ satisfies Eq. (3.16), $\mathbb {P}$-a.s..

Now we prove (3.18). By (3.26), there exists some constant $\varepsilon _0>0$ such that for any $\varepsilon \in (0,\varepsilon _0]$,

$$\begin{aligned} \mathbb {E}\left[ \left\| \frac{R_\varepsilon ^{u^\varepsilon }}{\sqrt{\varepsilon }h(\varepsilon )}\right\| ^2 \right] \le C(K,N,T,\alpha _0,\alpha _1,|p|,|q|). \end{aligned}$$

(3.30)

Notice that $b/\alpha$ is Lipschitz continuous and $\sigma /\alpha$ is bounded, then we have

$$\begin{aligned} \left| X_\varepsilon ^{u^\varepsilon } (t)\right| ^2&\le C(K,\alpha _0,\alpha _1)\int _0^t \left| X_\varepsilon ^{u^\varepsilon }(s)\right| ^2 ds+C(K,N,T,\alpha _0)\nonumber \\&\quad + \frac{C(K,\alpha _0)}{h^2(\varepsilon )}w^2(t)+C\left| \frac{R_\varepsilon ^{u^\varepsilon }(t)}{\sqrt{\varepsilon } h(\varepsilon )}\right| ^2. \end{aligned}$$

(3.31)

Hence by (1.6) and (3.30), for any $\varepsilon \in (0,\varepsilon _0]$, taking expectation in both sides in (3.31), we have

$$\begin{aligned} \mathbb {E}\left[ \left| X_\varepsilon ^{u^\varepsilon } (t)\right| ^2\right] \le C(K,\alpha _0,\alpha _1)\int _0^T \mathbb {E}\left[ \left| X_\varepsilon ^{u^\varepsilon }(s)\right| ^2\right] ds+C(K,N,T,\alpha _0,\alpha _1,|p|,|q|). \end{aligned}$$

By Gronwall’s inequality, we get

$$\begin{aligned} \mathbb {E}\left[ \left| X_\varepsilon ^{u^\varepsilon } (t)\right| ^2\right] \le C(K,N,T,\alpha _0,\alpha _1,|p|,|q|), \end{aligned}$$

(3.32)

then by Fubini’s theorem,

$$\begin{aligned} \mathbb {E}\left[ \int _0^T \left| X_\varepsilon ^{u^\varepsilon } (s)\right| ^2ds\right] \le C(K,N,T,\alpha _0,\alpha _1,|p|,|q|). \end{aligned}$$

(3.33)

First taking supremum with respect to $t\in [0,T]$ in (3.31), and then taking expectation in both sides, for any $\varepsilon \in (0,\varepsilon _0]$, by BDG inequality, (1.6), (3.30) and (3.33), we obtain that

$$\begin{aligned} \begin{aligned} \mathbb {E}\left[ \left\| X_\varepsilon ^{u^\varepsilon } \right\| ^2\right]&\le C(K,\alpha _0,\alpha _1)\mathbb {E}\left[ \int _0^T \left| X_\varepsilon ^{u^\varepsilon }(s)\right| ^2 ds\right] +C(K,N,T,\alpha _0,\alpha _1,|p|,|q|)\\&\le C(K,N,T,\alpha _0,\alpha _1,|p|,|q|), \end{aligned} \end{aligned}$$

which completes the proof. $\square$

Proposition 3.8

Under Hypothesis 2.1, for every fixed $N \in \mathbb {N}$, let $\{u^\varepsilon \}_{\varepsilon >0}$ be a family of processes in ${\mathcal {A}}_N$ that converges in distribution to some $u \in {\mathcal {A}}_N$ as $\varepsilon \rightarrow 0$, as random variables taking values in the space $S_N$, endowed with the weak topology. Then

$$\begin{aligned} \Gamma _{\varepsilon } \left( w(\cdot )+h(\varepsilon )\int _0^\cdot {\dot{u}}^\varepsilon (s) ds\right) \rightarrow \Gamma _0\left( \int _0^\cdot {\dot{u}}(s) ds \right) , \end{aligned}$$

in distribution in $C([0,T]; \mathbb {R}^d)$ as $\varepsilon \rightarrow 0$.

Proof

By the Skorokhod representation theorem, there exists a probability basis $({\bar{\Omega }},{\bar{{\mathcal {F}}}},({\bar{{\mathcal {F}}}}_t),{\bar{\mathbb {P}}})$, and on this basis, a Brownian motion ${{\bar{w}}}$ and a family of ${\bar{{\mathcal {F}}}}_t$-predictable processes $\{{{\bar{u}}}^\varepsilon \}_{\varepsilon >0}, {{\bar{u}}}$ taking values in $S_N$, ${\bar{\mathbb {P}}}$-a.s., such that the joint law of $(u^\varepsilon ,u, w)$ under $\mathbb {P}$ coincides with that of $({{\bar{u}}}^\varepsilon , {{\bar{u}}}, {{\bar{w}}})$ under ${\bar{\mathbb {P}}}$ and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\langle {{\bar{u}}}^\varepsilon -{{\bar{u}}}, g \rangle _{\mathcal {H}}=0, \ \ \forall g\in {\mathcal {H}}, \ {\bar{\mathbb {P}}}\text{- }a.s.. \ \ \end{aligned}$$

Let ${{\bar{X}}}_{\varepsilon }^{ {\bar{u}}^{\varepsilon }}$ be the solution of a similar equation to (3.16) with $u^\varepsilon$ replaced by ${{\bar{u}}}^\varepsilon$ and w by ${{\bar{w}}}$, and let ${{\bar{X}}}^{{{\bar{u}}}}$ be the solution of a similar equation to (3.9) with h replaced by ${{\bar{u}}}$. Thus, to prove this proposition, it is sufficient to prove that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \left\| {{\bar{X}}}_{\varepsilon }^{ {\bar{u}}^{\varepsilon }}-{{\bar{X}}}^{{{\bar{u}}}} \right\| =0, \quad \text { in probability}. \end{aligned}$$

(3.34)

From now on, we drop the bars in the notation for the sake of simplicity.

Notice that, for any $t \in [0,T]$,

$$\begin{aligned}&\quad X_\varepsilon ^{u^\varepsilon }(t) - X^u(t)\nonumber \\&=\int _0^t \left\{ \frac{1}{\sqrt{\varepsilon } h(\varepsilon )} \left[ \frac{b( q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s) )}{ \alpha (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} - \frac{b(q_{0} (s))}{\alpha (q_{0}(s))} \right] -D \left( \frac{b(q_0(s))}{\alpha (q_0(s))} \right) X^u(s) \right\} ds\nonumber \\&\quad + \int _0^t \left[ \frac{\sigma (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}{\alpha (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} {\dot{u}}^\varepsilon (s)-\frac{\sigma (q_0(s))}{\alpha (q_0(s))} {\dot{u}}(s) \right] ds\nonumber \\&\quad + \frac{1}{h(\varepsilon )} \int _0^t \frac{\sigma (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}{ \alpha (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} dw(s) +\frac{R_\varepsilon ^{u^\varepsilon }(t)}{\sqrt{\varepsilon } h(\varepsilon )}\nonumber \\&{=:} \sum _{k=1}^4 Y_\varepsilon ^{k,u^\varepsilon } (t). \end{aligned}$$

(3.35)

We shall prove this proposition in the following four steps.

Step 1: For the first term $Y_\varepsilon ^{1,u^\varepsilon }$, denote $x_\varepsilon (t):=\sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(t)$, by Taylor’s formula, there exists a random variable $\eta _\varepsilon$ taking values in (0, 1) such that

$$\begin{aligned} \begin{aligned}&\quad \left| Y_\varepsilon ^{1,u^\varepsilon } (t)\right| \\&= \left| \int _0^t \left[ D\left( \frac{b( q_{0} (s) + \eta _\varepsilon (s)x_\varepsilon (s) )}{ \alpha (q_{0} (s) + \eta _\varepsilon (s)x_\varepsilon (s))}\right) X_\varepsilon ^{u^\varepsilon }(s)-D \left( \frac{b(q_0(s))}{\alpha (q_0(s))} \right) X^u(s) \right] ds\right| \\&\le \left| \int _0^t D \left( \frac{b( q_{0} (s) + \eta _\varepsilon (s)x_\varepsilon (s) )}{ \alpha (q_{0} (s) + \eta _\varepsilon (s)x_\varepsilon (s))}\right) \cdot \left( X_\varepsilon ^{u^\varepsilon }(s)-X^u(s)\right) ds\right| \\&\quad +\left| \int _0^t \left[ D \left( \frac{b( q_{0} (s) + \eta _\varepsilon (s)x_\varepsilon (s) )}{ \alpha (q_{0} (s) + \eta _\varepsilon (s)x_\varepsilon (s))}\right) - D \left( \frac{b(q_0(s))}{\alpha (q_0(s))} \right) \right] \cdot X^u(s) ds\right| \\&{=:}y_\varepsilon ^{11}(t)+y_\varepsilon ^{12}(t). \end{aligned} \end{aligned}$$

For the first term $y_\varepsilon ^{11}$, by the boundness of $D\left( \frac{b}{\alpha }\right)$, we have

$$\begin{aligned} y_\varepsilon ^{11}(t) \le C(K,\alpha _0,\alpha _1) \int _0^t \left| X_\varepsilon ^{u^\varepsilon }(s)-X^u(s) \right| ds. \end{aligned}$$

(3.36)

Next we deal with the second term $y_\varepsilon ^{12}$. For each $R>\Vert q_0\Vert$ and $\rho \in (0,1)$, set

$$\begin{aligned} \eta _{R,\rho }:=\sup _{|x|\le R, |y|\le R, |x-y|\le \rho } \left| D\left( \frac{b}{\alpha }\right) (x)-D\left( \frac{b}{\alpha }\right) (y) \right| . \end{aligned}$$

Then by the continuous differentiability of $\frac{b}{\alpha }$, we know that for any fixed $R>0$,

$$\begin{aligned} \lim _{\rho \rightarrow 0}\eta _{R,\rho }=0. \end{aligned}$$

Since $\sqrt{\varepsilon } h(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0$, there exists some $\varepsilon _0>0$ small enough such that for all $0<\varepsilon \le \varepsilon _0$,

$$\begin{aligned} \sup _{\Vert q_0\Vert \le R, \sqrt{\varepsilon }h(\varepsilon ) \Vert X_{\varepsilon }^{u^\varepsilon }\Vert \le \rho }\left\| \left( D\left( \frac{b}{\alpha }\right) (q_0+\eta _{\varepsilon }\sqrt{\varepsilon }h(\varepsilon )X_{\varepsilon }^{u^\varepsilon } )-D\left( \frac{b}{\alpha }\right) (q_0) \right) X^u\right\| \le \eta _{R+1,\rho }\left\| X^u\right\| \end{aligned}$$

for any $\rho \in (0,1)$.

Thus, we obtain that for any $r>0, R>\Vert q_0\Vert$,

$$\begin{aligned}&\mathbb {P}\left( \left\| y_{\varepsilon }^{12}\right\|>r \right) \nonumber \\ \le&\mathbb {P}\left( \sqrt{\varepsilon }h(\varepsilon )\left\| X_{\varepsilon }^{u^\varepsilon } \right\|> \rho \right) + \mathbb {P}\left( \eta _{R+1,\rho } \left\| X^u\right\| >\frac{r}{T} \right) \nonumber \\ \le&\frac{\varepsilon h^2(\varepsilon )}{\rho ^2}\mathbb {E}\left[ \left\| X_{\varepsilon }^{u^\varepsilon }\right\| ^2\right] +\frac{T^2\eta _{R+1,\rho }^2}{r^2}\mathbb {E}\left[ \left\| X^u\right\| ^2\right] . \end{aligned}$$

(3.37)

By (3.10) and (3.19), letting $\varepsilon \rightarrow 0$ and then $\rho \rightarrow 0$ in (3.37), we can prove that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbb {P}\left( \left\| y_{\varepsilon }^{12}\right\|>r \right) =0, \ \ \ \ \text {for any } r>0. \end{aligned}$$

(3.38)

Step 2: For the second term $Y_\varepsilon ^{2,u^\varepsilon }$ we have

$$\begin{aligned} \begin{aligned}&\left| Y_\varepsilon ^{2,u^\varepsilon }(t)\right| \\ \le&\left| \int _0^t \frac{\sigma (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}{\alpha (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} \left( {\dot{u}}^\varepsilon (s)-{\dot{u}}(s)\right) ds\right| \\&+\left| \int _0^t \left[ \frac{\sigma (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}{\alpha (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} -\frac{\sigma (q_0(s))}{\alpha (q_0(s))}\right] {\dot{u}}(s) ds\right| \\ {=:}&\left| Y_\varepsilon ^{2,u^\varepsilon ,1}(t)\right| +\left| Y_\varepsilon ^{2,u^\varepsilon ,2}(t)\right| . \\ \end{aligned} \end{aligned}$$

Using the same argument as that in the proof of (3.14), we obtain that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\left\| Y_\varepsilon ^{2,u^\varepsilon ,1}\right\| =0, \ \text { a.s.}. \end{aligned}$$

(3.39)

Since $\left\| Y_\varepsilon ^{2,u^\varepsilon ,1}\right\| \le C(K,N,T,\alpha _0)$, by the dominated convergence theorem, Eq. (3.39) implies that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbb {E}\left\| Y_\varepsilon ^{2,u^\varepsilon ,1}\right\| = 0. \end{aligned}$$

Due to the Lipschitz continuity of $\sigma /\alpha$, we have

$$\begin{aligned} \begin{aligned} \left\| Y_\varepsilon ^{2,u^\varepsilon ,2}\right\| \le C(K,\alpha _0,\alpha _1)\int _0^T \sqrt{\varepsilon }h(\varepsilon )\left| X_\varepsilon ^{u^\varepsilon }(t)\right| \cdot \left| {\dot{u}}(t)\right| ds. \end{aligned} \end{aligned}$$

(3.40)

By (3.18) and Hölder’s inequality, we get

$$\begin{aligned} \mathbb {E}\left[ \int _0^T \left| X_\varepsilon ^{u^\varepsilon }(t)\right| \cdot \left| {\dot{u}}(t)\right| dt\right] \le C(K,N,T,\alpha _0,\alpha _1,\left| p\right| ,\left| q\right| ). \end{aligned}$$

Hence by (1.6), we obtain that

$$\begin{aligned} \mathbb {E}\left\| Y_\varepsilon ^{2,u^\varepsilon }\right\| \rightarrow 0, \ \ \text {as } \varepsilon \rightarrow 0. \end{aligned}$$

(3.41)

Step 3: For the third term $Y_\varepsilon ^{3,u^\varepsilon }$, by BDG inequality and (1.6), we have

$$\begin{aligned} \mathbb {E}\left\| Y_\varepsilon ^{3,u^\varepsilon }\right\|&=\frac{1}{h(\varepsilon )}\mathbb {E}\left[ \sup _{t\in [0,T]} \left| \int _0^t \frac{\sigma (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}{ \alpha (q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))} dw(s)\right| \right] \nonumber \\&\le \frac{C}{h(\varepsilon )} \mathbb {E}\left( \int _0^T\left\| \frac{(\sigma *\sigma ^T)(q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}{ \alpha ^2(q_{0} (s) + \sqrt{\varepsilon } h(\varepsilon )X_\varepsilon ^{u^\varepsilon }(s))}\right\| _{HS} ds\right) ^{\frac{1}{2}}\nonumber \\&\le \frac{C(K,T,\alpha _0) }{h(\varepsilon )} \rightarrow 0,\ \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

(3.42)

Step 4: For the last term $Y_\varepsilon ^{4,u^\varepsilon }$, by Lemma 3.7, we have

$$\begin{aligned} \mathbb {E}\left\| Y_\varepsilon ^{4,u^\varepsilon } \right\| \rightarrow 0, \ \ \ \text { as } \ \ \varepsilon \rightarrow 0. \end{aligned}$$

(3.43)

By Eq. (3.35) and (3.36), we obtain that

$$\begin{aligned}&\quad \sup _{0\le s\le t}\left| X_\varepsilon ^{u^\varepsilon }(s)-X^u(s)\right| \nonumber \\&\le C(K,\alpha _0,\alpha _1) \int _0^t \sup _{0\le v\le s}\left| X_\varepsilon ^{u^\varepsilon }(v)-X^u(v) \right| ds+ \sup _{0\le s\le t}y_\varepsilon ^{12}(s)\nonumber \\&\quad +\sup _{{0\le s\le t}}\left| Y_\varepsilon ^{2,u^\varepsilon }(s)\right| +\sup _{{0\le s\le t}}\left| Y_\varepsilon ^{3,u^\varepsilon }(s)\right| +\sup _{{0\le s\le t}}\left| Y_\varepsilon ^{4,u^\varepsilon }(s)\right| . \end{aligned}$$

(3.44)

Using Gronwall’s inequality, we have that

$$\begin{aligned} \left\| X_\varepsilon ^{u^\varepsilon }-X^u \right\| \le C\left( \left\| y_\varepsilon ^{12}\right\| +\sum _{l=2,3,4} \left\| Y_\varepsilon ^{l,u^\varepsilon }\right\| \right) . \end{aligned}$$

This, together with (3.38), (3.41), (3.42) and (3.43), implies that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \left\| X_\varepsilon ^{u^\varepsilon }-X^u\right\| =0, \quad \text {in probability}, \end{aligned}$$

which completes the proof. $\square$

According to Theorem 3.1, the MDP of $\{X_\varepsilon ^M\}_{\varepsilon >0}$ follows from Proposition 3.3 and Proposition 3.8, which completes the proof of our main result Theorem 2.2.