Moderate Deviation Principle for Multiscale Systems Driven by Fractional Brownian Motion

Abstract

In this paper, we study the moderate deviations principle (MDP) for slow–fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm) with Hurst parameter $H\in (1/2,1)$. We derive conditions on the moderate deviations scaling and on the Hurst parameter H under which the MDP holds. In addition, we show that in typical situations the resulting action functional is discontinuous in H at $H=1/2$, suggesting that the tail behavior of stochastic dynamical systems perturbed by fBm can have different characteristics than the tail behavior of such systems that are perturbed by standard Brownian motion.

Appendices

Appendix

Appendix A. Fractional Brownian Motion and Pathwise Stochastic Integration

1.1 A.1 Fractional Brownian Motion: Definition and Main Properties

A one-dimensional fractional Brownian motion (fBm) is a centered Gaussian process $W^H = \left\{ W^H_t :t \in [0,1] \right\} \subset L^2(\Omega )$, characterized by its covariance function

$$\begin{aligned} R_H(t,s) = E (W^H_t W^H_s) = \frac{1}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right) . \end{aligned}$$

It is straightforward to verify that increments of fBm are stationary. The parameter $H \in (0,1)$ is usually referred to as the Hurst exponent, Hurst parameter, or Hurst index.

By Kolmogorov’s continuity criterion, such a process admits a modification with continuous sample paths, and we always choose to work with such. In this case, one may show in fact that almost every sample path is locally Hölder continuous of any order strictly less than H. It is this sense in which it is often said that the value of H determines the regularity of the sample paths.

Note that when $H = \frac{1}{2}$, the covariance function is $R_{\frac{1}{2}}(t,s) = t \wedge s$. Thus, one sees that $W^{\frac{1}{2}}$ is a standard Brownian motion, and in particular that its disjoint increments are independent. In contrast to this, when $H \ne \frac{1}{2}$, nontrivial increments are not independent. In particular, when $H > \frac{1}{2}$, the process exhibits long-range dependence.

Note moreover that when $H \ne \frac{1}{2}$, the fractional Brownian motion is not a semimartingale, and the usual Itô calculus therefore does not apply.

Another noteworthy property of fractional Brownian motion is that it is self-similar in the sense that, for any constant $a >0$, the processes $\left\{ W^H_t :t \in [0,1]\right\}$ and $\left\{ a^{-H} W^H_{at}:t \in [0,1]\right\}$ have the same distribution.

Finally, an n-dimensional fractional Brownian motion is a random vector where the components are independent one-dimensional fractional Brownian motions with the same Hurst parameter $H\in (0,1)$.

The self-similarity and long-memory properties of the fractional Brownian motion make it an interesting and suitable input noise in many models in various fields such as analysis of financial time series, hydrology, and telecommunications. However, in order to develop interesting models based on fractional Brownian motion, one needs an integration theory with respect to it, which we present in the next subsection.

1.2 A.2 Pathwise Stochastic Integration with Respect to Fractional Brownian Motion

Stochastic integrals with respect to fractional Brownian motion can be understood, when $H \ge 1/2$, as generalized Stieltjes integral as introduced in the work of Zähle [51]. Let $f \in L^1([a,b])$ and $\alpha > 0$. The left-sided and right-sided fractional Riemann–Liouville integrals of f of order $\alpha$ are defined for almost all $x \in [a,b]$ by

$$\begin{aligned} I_{a^+}^{\alpha }f(x) = \frac{1}{\Gamma (\alpha )}\int _a^x (x-y)^{\alpha -1}f(y)dy \end{aligned}$$

and

$$\begin{aligned} I_{b^-}^{\alpha }f(x) = \frac{1}{\Gamma (\alpha )}\int _x^b (y-x)^{\alpha -1}f(y)dy, \end{aligned}$$

respectively, where $\Gamma (\alpha )$ is the Euler gamma function. This naturally leads to the definition of the function spaces

$$\begin{aligned} I_{a^+}^{\alpha }(L^p([a,b]))=\{g=I_{a^+}^{\alpha }(f):f\in L^p([a,b]) \} \end{aligned}$$

and

$$\begin{aligned} I_{b^-}^{\alpha }(L^p([a,b]))=\{g=I_{b^-}^{\alpha }(f):f\in L^p([a,b]) \}. \end{aligned}$$

The following integration by parts formula holds

$$\begin{aligned} \int _a^b I_{a^+}^{\alpha }f(x) g(x)dx=\int _a^b f(x) I_{b^-}^{\alpha }g(x)dx \end{aligned}$$

for $f\in L^p([a,b]),g\in L^q([a,b])$ such that $1/p+1/q\le 1+\alpha$.

For $0< \alpha <1$, we can define the fractional derivatives

$$\begin{aligned} D^\alpha _{a^+}f(x)=\frac{d}{dx}I^{1-\alpha }_{a^+} f(x)=\frac{1}{\Gamma (1-\alpha )}\frac{d}{dx}\int _a^x (x-t)^{-\alpha }f(t)dt \end{aligned}$$

and

$$\begin{aligned} D^\alpha _{b^-}f(x)=\frac{d}{dx}I^{1-\alpha }_{b^-} f(x)=\frac{1}{\Gamma (1-\alpha )}\frac{d}{dx}\int _x^b (t-x)^{-\alpha }f(t)dt \end{aligned}$$

as long as the right-hand sides are well-defined. Furthermore, if $f \in I_{a^+}^{\alpha }\left( L^p([a,b])\right)$ (respectively $f \in I_{b^-}^{\alpha }(L^p([a,b]))$ and $0< \alpha <1$, then the previous fractional derivatives admit the Weyl representation

$$\begin{aligned} D_{a^+}^{\alpha }f(x) = \frac{1}{\Gamma (1-\alpha )} \left( \frac{f(x)}{(x-a)^{\alpha }} + \alpha \int _a^x \frac{f(x) - f(y)}{(x-y)^{\alpha -1}}dy \right) \mathbbm {1}_{(a,b)}(x) \end{aligned}$$

and

$$\begin{aligned} D_{b^-}^{\alpha }f(x) = \frac{1}{\Gamma (1-\alpha )} \left( \frac{f(x)}{(b-x)^{\alpha }} + \alpha \int _x^b \frac{f(x) - f(y)}{(y-x)^{\alpha -1}}dy \right) \mathbbm {1}_{(a,b)}(x), \end{aligned}$$

respectively, for almost all $x \in [a,b]$. There is also the integration by parts formula

$$\begin{aligned} \int _a^b D_{a^+}^{\alpha }f(x) g(x)dx=\int _a^b f(x) D_{b^-}^{\alpha }g(x)dx \end{aligned}$$

for $f \in I_{a^+}^{\alpha }(L^p([a,b])),g \in I_{b^-}^{\alpha }(L^q([a,b]))$ such that $1/p+1/q\le 1+\alpha$.

The upcoming lemma contains a useful technical result in [41].

Lemma 5

Let $p\ge 1$ and $b>0$. Then, the operator $t^\beta I_{0^+}^\alpha t^\gamma$ is bounded in $L^p([0,b])$ if $\alpha >0, \alpha +\beta +\gamma =0$ and $(\gamma +1)p>1$. Meanwhile, the operator $t^\beta I_{1^-}^\alpha t^\gamma$ is bounded in $L^p([0,b])$ if $\alpha >0, \alpha +\beta +\gamma =0$ and $(\alpha +\gamma )p<1$.

Proof

This is a consequence of Samko et al. [41, (5.45’) and (5.46’)]. $\square$

We refer to Samko et al. [41] for more detailed properties of fractional operators.

Next, let $f(a+) = \lim _{x \searrow 0} f(a + x)$ and $g(b-) = \lim _{x \searrow 0} g(b-x)$ and define

$$\begin{aligned}&f_{a^+}(x) = (f(x) - f(a+))\mathbbm {1}_{(a,b)}(x) \\&g_{b^-}(x) = (g(x) - g(b-))\mathbbm {1}_{(a,b)}(x). \end{aligned}$$

We recall from Zähle et al. [51] the definition of generalized Stieltjes fractional integrals with respect to irregular functions (in the sense of which we view the stochastic integrals with respect to fractional Brownian motion appearing in this paper).

Definition 2

(Generalized Stieltjes integral) Suppose that f and g are functions such that $f(a+)$, $g(a+)$ and $g(b-)$ exist, $f_{a^+} \in I_{a^+}^{\alpha }(L^p([a,b]))$ and $g_{b^-} \in I_{b^-}^{1-\alpha }(L^p([a,b]))$ for some $p,q \ge 1$, $1/p + 1/q \le 1$, $0< \alpha <1$. Then, the integral of f with respect to g is defined by

$$\begin{aligned} \int _a^b f dg = (-1)^{\alpha } \int _a^b D_{a^+}^{\alpha }f(x)D_{b^-}^{1-\alpha }g_{b^-}(x)dx + f(a+)\left( g(b-) - g(a+) \right) . \end{aligned}$$

Remark 11

If $\alpha p <1$, under the assumptions of the preceding definition, we have that $f \in I_{a^+}^{\alpha }(L^p([a,b]))$ and we can write

$$\begin{aligned} \int _a^b f dg = (-1)^{\alpha } \int _a^b D_{a^+}^{\alpha }f_{a^+}(x)D_{b^-}^{1-\alpha }g_{b^-}(x)dx. \end{aligned}$$

In [51], it was further shown that if f and g are, respectively, $\lambda$ and $\mu$-Hölder continuous such that $\lambda +\mu >1$, then the conditions for the generalized Stieltjes integral $\int _a^b f dg$ are satisfied for $p=q=\infty$ and $\alpha<\lambda ,1-\alpha <\mu$. In particular, this class of generalized Stieltjes integrals with Hölder continuous f, g coincides with the class of Riemann–Stieltjes integrals studied in [49] by Young. We note here that any Young’s integrals appearing in this paper are constructed from Hölder continuous paths of a fractional Brownian motion. Further details are given in [51, Sect. 5.1].

1.3 A.3 The Cameron–Martin Space of Fractional Brownian Motion

Consider the deterministic kernel

$$\begin{aligned} K_H(t,s)=c_Hs^{1/2-H}\left( \int _s^t (u-s)^{H-3/2}u^{H-1/2}du\right) \mathbbm {1}_{\{t>s\}} \end{aligned}$$

for which $c_H=\left( H(2H-1)/\beta (2-2H,H-1/2)\right) ^{1/2}$. Slightly abusing notation, we also write $K_H$ for the integral operator

$$\begin{aligned} K_Hg (s)=\int _0^s K_H(s,r)g(r)dr. \end{aligned}$$

For $H \ge 1/2$, the operator $K_H$ can be represented as

$$\begin{aligned} K_Hg=c_H\Gamma (H-1/2)I^1_{0^+}t^{H-1/2}I^{H-1/2}_{0^+}t^{1/2-H}g. \end{aligned}$$

Additionally, we denote by ${\dot{K}}_H$ the “derivation” of the operator $K_H$, i.e.,

$$\begin{aligned} {\dot{K}}_Hg = c_H\Gamma (H-1/2)t^{H-1/2}I^{H-1/2}_{0^+}t^{1/2-H}g. \end{aligned}$$

(41)

The Cameron–Martin space ${\mathcal {H}}_H$ associated with $W^H$ is

$$\begin{aligned} {\mathcal {H}}_H=\{K_H{\hat{g}}:{\hat{g}}\in L^2\left( [0,1];{\mathbb {R}}^m\right) \}, \end{aligned}$$

equipped with the inner product $\left\langle g,f \right\rangle _{{\mathcal {H}}_H}=\left\langle {\hat{g}},{\hat{f}} \right\rangle _{L^2\left( [0,1];{\mathbb {R}}^m\right) }$. Note that later on, we will alternate between ${\hat{g}}$ and $K_H^{-1}g$, which are equivalent ways of writing the same quantity.

In this paper, the noise process we consider in the slow–fast systems we study is of the form

$$\begin{aligned} \left\{ \left( W^H(t),B(t)\right) :t\in [0,1]\right\} , \end{aligned}$$

where B is a m-dimensional standard Brownian motion, $W^H$ is a p-dimensional fractional Brownian motion of Hurst parameter H and they are independent. We will hence need to work with the Cameron–Martin space associated with the process $(W^H,B)$, which, based on the previous description, is defined to be the space ${\mathcal {S}}$ given by

$$\begin{aligned} \left\{ \left( K_H{\hat{g}}_1,K_{1/2}{\hat{g}}_2\right) :\left( {\hat{g}}_1,{\hat{g}}_2\right) \in L^2\left( [0,1];{\mathbb {R}}^{m+p}\right) \right\} . \end{aligned}$$

(42)

As a Cameron–Martin space, ${\mathcal {S}}$ is a Hilbert space equipped with the inner product given by

$$\begin{aligned} \left\langle \left( g_1,g_2\right) ,\left( f_1,f_2\right) \right\rangle _{{\mathcal {S}}}=\left\langle g_1,f_1 \right\rangle _{{\mathcal {H}}_H}+\left\langle g_2,f_2 \right\rangle _{{\mathcal {H}}_{1/2}}. \end{aligned}$$

Let us now state an important fact regarding the differentiability of elements in ${\mathcal {H}}_H$ when $H> 1/2$ which we will need throughout the paper.

Lemma 6

If $H> 1/2$ and $u\in {\mathcal {H}}_H$ such that $u=K_H {\hat{u}},{\hat{u}}\in L^2([0,1];{\mathbb {R}}^n)$, then we have

$$\begin{aligned} {\dot{u}}(t)={\dot{K}}_H{\hat{u}}(t)&=c_H\Gamma (H-1/2)t^{H-1/2}I^{H-1/2}_{0^+}t^{1/2-H}{\hat{u}}(t)\\&=c_H t^{H-1/2}\int _0^t (t-s)^{H-3/2}s^{1/2-H}{\hat{u}}_sds, \end{aligned}$$

such that $c_H=\left( H(2H-1)/\beta (2-2H,H-1/2)\right) ^{1/2}$.

Meanwhile, if $H=1/2$ and $u\in {\mathcal {H}}_{1/2}$, then ${\dot{u}}_t={\hat{u}}_t$.

Proof

This is a direct consequence of formula (41). $\square$

The following is another important property of the operator ${\dot{K}}_H$.

Proposition 10

The map ${\dot{K}}_H$ as described in Lemma 6 is a bounded operator in $L^2([0,1];{\mathbb {R}}^n)$.

Proof

The assumptions of Lemma 5 are satisfied for $p=2,\alpha =H-1/2,\beta =0$ and $\gamma =1/2-H$; hence, the operator $I^{H-1/2}_{0^+}t^{1/2-H}$ is bounded in $L^2([0,1];{\mathbb {R}}^n)$. Since ${\dot{K}}_H=t^{H-1/2}I^{H-1/2}_{0^+}t^{1/2-H}$ based on Lemma 6, this implies

$$\begin{aligned} \left\Vert {\dot{K}}_H f \right\Vert _{L^2([0,1];{\mathbb {R}}^n)}&=\left\Vert t^{H-1/2}I^{H-1/2}_{0^+}t^{1/2-H}f \right\Vert _{L^2([0,1];{\mathbb {R}}^n)}\le \left\Vert I^{H-1/2}_{0^+}t^{1/2-H}f \right\Vert _{L^2([0,1];{\mathbb {R}}^n)}\\&\le C\left\Vert f \right\Vert _{L^2([0,1];{\mathbb {R}}^n)}. \end{aligned}$$

$\square$

For more details about fractional Brownian motion, we refer the reader to the monographs [4, 36].

1.4 A.4 Results Related to Young’s Integrals

The two results presented here provide us with a way of bounding Young’s integrals and with a version of change of variable formula for differential equations that contain Young’s integrals, respectively.

Lemma 7

(Young–Loéve’s inequality) Let f and g be, respectively, $\alpha$ and $\beta$-Hölder continuous, such that $\alpha +\beta >1$. Then, a.s one has

$$\begin{aligned} \left|\int _r^t f_sdg_s-f_r(g_t-g_r) \right|\le C \left|f \right|_\alpha \left|g \right|_\beta \left|t-r \right|^{\alpha +\beta }. \end{aligned}$$

Moreover, assume f is bounded, then

$$\begin{aligned} \left|\int _r^t f_sdg_s \right|\le C\left|f \right|_\alpha \left|g \right|_\beta \left|t-r \right|^{\alpha +\beta }+\left|f \right|_\infty \left|t-r \right|^\beta \le C\left|f \right|_\alpha \left|g \right|_\beta \left|t-r \right|^\beta . \end{aligned}$$

Proof

Refer to Friz and Victoir [22, Proposition 6.4]. $\square$

Theorem 3

For $i=1,\ldots ,m$, let $0<\alpha _i<1/2$, $f^i\in I^{\alpha _i}_{0+}(L^2([0,b]))$ be bounded and $g^i_{b-}\in I^{1-\alpha _i}_{b-}(L^2([0,b]))$, where the function $g^i_{b-}$ is defined in Lemma 5. Moreover, assume $h=(h^1,\ldots ,h^m)$ such that

$$\begin{aligned} h^i_t=h^i_0+\int _0^t f^i_sdg^i_s. \end{aligned}$$

Then, for any ${\mathcal {C}}^1$ mapping $F:{\mathbb {R}}^m\times {\mathbb {R}}\rightarrow {\mathbb {R}}^n$ such that $\frac{\partial F}{\partial x_i}\in {\mathcal {C}}^1,i=1,\ldots ,m$ and $r\le t\le T$, it holds that

$$\begin{aligned} F(h_t,t)-F(h_r,r)=\sum _{i=1}^m\int _r^t \frac{\partial F}{\partial x_i}(h_s,s)f^i_sdg^i_s +\int _r^t \frac{\partial F}{\partial s}(h_s,s) ds. \end{aligned}$$

In particular, this change of variable formula applies to the special case when $f_i$ and $g_i$ are, respectively, $\lambda _i$ and $\mu _i$-Hölder continuous such that $\lambda _i+\mu _i>1,i=1,\ldots ,m$.

Proof

For the change of variable formula in the general case, we refer to Zähle [50, Theorem 5.2].

Now, let us consider the special case and assume there is a constant C such that $\left|f_i \right|_{\lambda _i}$,$\left|g_i \right|_{\mu _i}<C$ and $\lambda _i+\mu _i>1$ for $1\le i\le m$. Then, one can choose $\alpha _i$ in the interval (0, 1/2) such that $\lambda _i>\alpha _i$ and $\mu _i>1-\alpha _i$ for $1\le i\le m$. Based on the previous fact, Lemmas 13.2 and 13.2’ in [41] imply, respectively, that

$$\begin{aligned} f^i\in I^{\alpha _i}_{0+}(L^2([0,b])),\qquad g^i_{b-}\in I^{1-\alpha _i}_{b-}(L^2([0,b])). \end{aligned}$$

Moreover, $f^i$ as Hölder continuous functions on [0, b] are necessarily bounded. Consequently, the general change of variable formula covers this particular case. $\square$

Appendix B. Regularity Results and Other Technical Lemmas

This appendix gathers results related to Poisson equations as well as the technical lemmas required for the analysis of the control problems.

1.1 B.1 Results Related to Poisson Equations

The following theorem is a consequence of Pardoux and Veretennikov [37, Theorem 2] and [38, Theorem 3] for solutions of Poisson equations. Let ${\mathcal {L}}$ be the infinitesimal generator defined in (5).

Theorem 4

Recall ${\mathcal {C}}^{2,\zeta }({\mathbb {R}}^n\times {\mathcal {Y}})$ for some $\zeta >0$ is the function space defined at the beginning of Sect. 2. Let $h \in {\mathcal {C}}^{2,\zeta }({\mathbb {R}}^n\times {\mathcal {Y}})$ such that

$$\begin{aligned} \int _{{\mathcal {Y}}} h(x,y)\mu (dy)=0 \end{aligned}$$

and that for some positive constants K and $D_h$,

$$\begin{aligned} \left|h(x,y) \right| +\left|\nabla _x h(x,y) \right|+ \left|\nabla ^2_x h(x,y) \right|\le K\left( 1+\left|y \right|^{D_h}\right) \end{aligned}$$

uniformly with respect to x.

Then, there is a unique solution to

$$\begin{aligned} {\mathcal {L}} u(x,y)=-h(x,y),\quad \int _{\mathcal {Y}} u(x,y) \mu (dy)=0. \end{aligned}$$

Moreover, $u(\cdot ,y)\in {\mathcal {C}}^2$, $\nabla ^2_x u\in {\mathcal {C}}({\mathbb {R}}^n\times {\mathcal {Y}})$ and there exists a positive constant M such that

$$\begin{aligned} \left|u(x,y) \right|+\left|\nabla _y u(x,y) \right|+\left|\nabla _x u(x,y) \right|+\left|\nabla ^2_x u(x,y) \right|+\left|\nabla _y\nabla _x u(x,y) \right|\le M(1+\left|y \right|^{D_h}). \end{aligned}$$

Remark 12

Consider the Poisson equation in (6). Under Conditions H1 and H2-A, Theorem 4 states that there exists a positive constant C such that, uniformly,

$$\begin{aligned} \left|\phi (x,y) \right|+\left|\nabla _y \phi (x,y) \right|+\left|\nabla _x \phi (x,y) \right|+\left|\nabla ^2_x \phi (x,y) \right|+\left|\nabla _y\nabla _x \phi (x,y) \right|< C. \end{aligned}$$

(43)

On the other hand, under Conditions H1 and H2-B, Theorem 4 states that there exists a positive constant C such that, uniformly with respect to x,

$$\begin{aligned} \left|\phi (x,y) \right|+\left|\nabla _y \phi (x,y) \right|+\left|\nabla _x \phi (x,y) \right|+\left|\nabla ^2_x \phi (x,y) \right|+\left|\nabla _y\nabla _x \phi (x,y) \right|< C\left( 1+\left|y \right|^{D_g}\right) . \end{aligned}$$

(44)

1.2 B.2 Ancillary Results Related to the Control Problems

This subsection gathers all technical results related to the study of the control problems appearing throughout the paper.

Lemma 8

(Lemma 5.2 in [48]) Let $H,H'$ be Hilbert spaces and $a:H \rightarrow H'$ be a bounded linear operator. Moreover, let $q=aa^*$ and $q^{-1}$ be the inverse of q. Then, for any $u\in H$,

$$\begin{aligned} \left\Vert a^*q^{-1}a u \right\Vert _{H}\le \left\Vert u \right\Vert _{H}. \end{aligned}$$

Lemma 9

Assume that for all x and nonzero $z\in {\mathbb {R}}^n$,

$$\begin{aligned} \left\langle \int _{{\mathcal {Y}}}\nabla _y\phi \left( x,y\right) \sigma \left( y\right) \left( \nabla _y\phi \left( x,y\right) \sigma \left( y\right) \right) ^\top \mu (dy)z,z \right\rangle >0. \end{aligned}$$

Then, the operator $Q^{H}_x$ defined in (36) is invertible and its inverse $(Q^{H}_x)^{-1}$ is a bounded in $L^2\left( [0,1];{\mathbb {R}}^n\right)$.

Proof

Using the operators $\pi ,\pi ^*,\rho ,\rho ^*$ defined in Section 5.5, we have $Q^{H}_xh=\left( \pi \pi ^*+\rho \rho ^*\right) h$ with

$$\begin{aligned} \rho \rho ^*h(t)=\int _{{\mathcal {Y}}} {\nabla _y\phi \left( x,y\right) \sigma \left( y\right) }\left( \nabla _y\phi \left( x,y\right) \sigma \left( y\right) \right) ^\top h(t,y)\mu (dy). \end{aligned}$$

Furthermore, $\pi \pi ^*,\rho \rho ^*$ are positive and self-adjoint operators, which means that $Q^{H}_x$ is also positive and self-adjoint. In addition, the fact that $Q^{H}_{x}\ge \rho \rho ^*$ and Condition H2-A or H2-B imply that $(Q^{H}_{x})^2\ge \left( \rho \rho ^*\right) ^2>0$. This leads to

$$\begin{aligned} \inf _{\left\Vert h \right\Vert _{L^2\left( [0,1];{\mathbb {R}}^n\right) }=1}\left\Vert Q^{H}_xh \right\Vert _{L^2\left( [0,1];{\mathbb {R}}^n\right) }&=\inf _{\left\Vert h \right\Vert _{L^2\left( [0,1];{\mathbb {R}}^n\right) }=1}\left\langle (Q^{H}_x)^2 h,h \right\rangle _{L^2\left( [0,1];{\mathbb {R}}^n\right) }\\&\ge \inf _{\left\Vert h \right\Vert _{L^2\left( [0,1];{\mathbb {R}}^n\right) }=1}\left\langle \left( \rho \rho ^*\right) ^2h,h \right\rangle _{L^2\left( [0,1];{\mathbb {R}}^n\right) }>0, \end{aligned}$$

so that $Q^{H}_x$ is bounded from below and $\ker Q^{H}_x=\{0\}$. This combined with self-adjointness implies

$$\begin{aligned} {\text {Im}} Q^{H}_{x}=\left[ \ker \left( Q^{H}_{x}\right) ^*\right] ^\bot =\left[ \ker Q^{H}_{x}\right] ^\bot =\{0\}^\bot =L^2\left( [0,1];{\mathbb {R}}^n\right) . \end{aligned}$$

It follows that $Q^{H}_{{\bar{X}}}$ is bijective. The operator $Q^{H}_{{\bar{X}}}$ is also bounded in $L^2\left( [0,1];{\mathbb {R}}^n\right)$ via Proposition 10, so we can conclude it has a bounded inverse by the inverse mapping theorem. $\square$

Lemma 10

It can be assumed that there exists a finite constant N such that, almost surely, the control process $w^{\epsilon }$ appearing in the variational representation (12) satisfies

$$\begin{aligned} \sup _{\epsilon >0}\left\Vert w^\epsilon \right\Vert ^2_{{\mathcal {S}}}\le N. \end{aligned}$$

Proof

This is an immediate consequence of Zhang [52, Theorem 3.2]. $\square$

Lemma 11

Assume $w^\epsilon \in {\mathcal {S}}$ is a control such that

$$\begin{aligned} \sup _{\epsilon>0}\left\Vert w^\epsilon \right\Vert ^2_{{\mathcal {S}}}=\sup _{\epsilon >0}\int _0^1 \left|{\hat{u}}^\epsilon _s \right|^2+\left|{\hat{v}}^\epsilon _s \right|^2 ds< N \end{aligned}$$

for some finite constant N. Then, under Condition H1, it holds that for $\epsilon _0>0$ small enough,

$$\begin{aligned} \sup _{\epsilon<\epsilon _0}{\mathbb {E}}{\left[ \int _0^1\left|Y^{\epsilon ,w^\epsilon }_s \right|^2ds\right] }<C \end{aligned}$$

for some constant $C>0$, which further implies that

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{t\in [0,1]}\left|Y^{\epsilon ,w^\epsilon }_t \right|\right] }\le \frac{C}{\sqrt{\epsilon }}. \end{aligned}$$

Proof

The first estimate was proven in [47, Lemma 3.1]. For the second estimate, the dissipative property of the drift coefficient of $Y^{\epsilon ,w^\epsilon }$ and Itô’s formula yield

$$\begin{aligned} Y^{\epsilon ,w^\epsilon }_t&=e^{-\frac{1}{\epsilon }\Gamma t}y_0+\int _0^t \frac{1}{\epsilon } e^{-\frac{1}{\epsilon }(t-s)}\zeta (Y^{\epsilon ,w^\epsilon })ds+\int _0^t\frac{h(\epsilon )}{\sqrt{\epsilon }}e^{-\frac{1}{\epsilon }(t-s)}\sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{v}}^\epsilon _s ds\\&\quad +\,\int _0^t\frac{1}{\sqrt{\epsilon }}e^{-\frac{1}{\epsilon }(t-s)}\sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) dB_s. \end{aligned}$$

We then apply the Burkhölder–Davis–Gundy inequality to the Itô integral term and Hölder’s inequality to the Riemann integral terms to get

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{t\in [0,1]}\left|Y^{\epsilon ,w^\epsilon }_t \right|\right] }\\ {}&\quad \le \sup _{t\in [0,1]}e^{-\frac{1}{\epsilon }\Gamma t}y_0+\frac{1}{\epsilon }\sqrt{\int _0^t e^{-\frac{2}{\epsilon }\Gamma (t-s)} ds}\sqrt{{\mathbb {E}}{\left[ \int _0^1 \left|\zeta (Y^{\epsilon ,w^\epsilon }) \right|^2 ds \right] }}\\&\qquad +\frac{h(\epsilon )}{\sqrt{\epsilon }}\sqrt{\int _0^t e^{-\frac{2}{\epsilon }\Gamma (t-s)} ds\int _0^1 \left|{\dot{v}}^\epsilon _s \right|^2 ds }+\frac{1}{\sqrt{\epsilon }}\sqrt{{\mathbb {E}}{\left[ \int _0^1 e^{-\frac{2}{\epsilon }\Gamma (t-s)}\left|\sigma (Y^{\epsilon ,w^\epsilon }_s) \right|^2 ds\right] }}. \end{aligned}$$

Since $\sigma (y)\sigma ^T(y)$ is bounded and $\zeta (y)$ is sublinear, the first estimate of this lemma can be applied to the expression ${\mathbb {E}}{\left[ \int _0^1 \left|\zeta (Y^{\epsilon ,w^\epsilon }) \right|^2 ds \right] }$. Then, the simple fact that $\int _r^t e^{-\frac{2}{\epsilon }\Gamma (t-s)}ds\le \epsilon \int _0^\infty e^{-2\Gamma s}ds=\frac{\epsilon }{2\Gamma }$ implies that

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{t\in [0,1]}\left|Y^{\epsilon ,w^\epsilon }_t \right|\right] }\le C\left( \frac{1}{\sqrt{\epsilon }}+h(\epsilon )\right) \le \frac{C}{\sqrt{\epsilon }}. \end{aligned}$$

$\square$

Lemma 12

Assume $w^\epsilon \in {\mathcal {S}}$ is a control such that

$$\begin{aligned} \sup _{\epsilon>0}\left\Vert w^\epsilon \right\Vert ^2_{{\mathcal {S}}}=\sup _{\epsilon >0}\int _0^1 \left|{\hat{u}}^\epsilon _s \right|^2+\left|{\hat{v}}^\epsilon _s \right|^2 ds< N \end{aligned}$$

for some finite constant N.

• (i) Under Conditions H1 and H2-A, there exist constants C that change from line to line such that

  $$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) g\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s \right) ds \right|\right] }\le C\rho ,\\&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left|\int _r^t\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) dB_s \right|^2\right] }\le C\rho \\&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{v}}^\epsilon _sds \right|^2\right] }\le C\rho \\&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _sds \right|^2\right] }\le C\rho ,\\&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}}\left|\int _0^t f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _sds \right|^2\right] }\le C\rho . \end{aligned}$$

• (ii) Under Conditions H1 and H2-B, there exist constants C that change from line to line such that for any q in $\Big (1,\frac{1}{D_f+D_g}\Big ]$, we have

  $$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) g\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s \right) ds \right|^q\right] }\le C\rho ^{q-1},\\&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left|\int _r^t\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) dB_s \right|^{2q}\right] }\le C\rho ^{q-1}\\&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{v}}^\epsilon _sds \right|^{2q}\right] }\le C\rho ^{q-1},\\&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _sds \right|^{2q}\right] }\le C\rho ^{q-1},\\&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}}\left|\int _0^t f\left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _sds \right|^{2q}\right] }\le C\rho ^{q-1}. \end{aligned}$$

Proof

We start with part (i). The first estimate is straightforward due to the boundedness of $\nabla _x\phi (x,y)$ stated in (43) and the boundedness of g(x, y) guaranteed by Condition H2-A. For the second estimate, we assume that $0\le r\le t\le 1$ and apply the Burkhölder–Davis–Gundy inequality to obtain

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r\le t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) dB_s \right|^2\right] }\\&\quad \le {\mathbb {E}}{\left[ \left( \int _r^{r+\rho }\left|\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) \right|^2 ds\right) \right] }\le C\rho . \end{aligned}$$

For the third estimate, we can write

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{v}}^\epsilon _sds \right|^2\right] }\\&\quad \le {\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left( \int _r^t \left|\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) \right|^2 ds\int _0^1 \left|{\dot{v}}^\epsilon _s \right|^2ds\right) \right] }\\&\quad \le C{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left( \int _r^t \left|\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) \right|^2 ds\right) \right] } \le C \rho . \end{aligned}$$

The last inequality in part (i) is a consequence of the boundedness of $\sigma (y)\sigma ^\top (y)$ in Condition H1 and the boundedness of $\nabla _x\phi (x,y)$ stated in (43) (requiring Condition H2-A). Finally, the two remaining estimates of part (i) are derived similarly to the previous one.

We continue with part (ii). For the first inequality, the sublinear growth of $\nabla _x\phi (x,y)$ in y stated at (44) (requiring Condition H2-B) and the sublinear growth of g(x, y) in y from Condition H2-B imply for any q in $\Big (1,\frac{1}{D_g}\Big ]$,

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) g\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s \right) ds \right|^q\right] }\\&\quad \le C{\mathbb {E}}{\left[ \left( \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} { \int _r^t {1+\left|Y^{\epsilon ,w^\epsilon }_s \right|^{2D_g}} ds}\right) ^q\right] }\\&\quad \le C\rho ^{q-1}{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} { \int _r^t \left( 1+\left|Y^{\epsilon ,w^\epsilon }_s \right|^{2qD_g}\right) ds}\right] }\le C\rho ^{q-1}, \end{aligned}$$

where the last inequality is due to Lemma 11. For the second estimate, assume that $0\le r\le t\le 1$. Then, the Burkhölder–Davis–Gundy inequality combined with the sublinear growth of $\nabla _y\phi (x,y)$ in y (requiring Condition H2-B) and the boundedness of $\sigma (y)\sigma ^\top (y)$ in Condition H1 imply that for any q in $\Big (1,\frac{1}{D_g}\Big ]$,

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r\le t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) dB_s \right|^{2q}\right] }\\&\quad \le {\mathbb {E}}{\left[ \left( \int _r^{r+\rho }\left|\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) \right|^2 ds\right) ^{q}\right] }\\&\quad \le C\rho ^{q-1}{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} { \int _r^t {1+\left|Y^{\epsilon ,w^\epsilon }_s \right|^{2qD_g}} ds}\right] }\le C\rho ^{q-1}. \end{aligned}$$

The arguments for the three remaining estimates of part (ii) are similar, so we will handle one case only. The sublinear growth of $\nabla _x\phi (x,y)$ in y stated at (44) (requiring Condition H2-B) and sublinear growth of f(y) in y in Condition H2-B imply that for any q in $\Big (1,\frac{1}{D_f+D_g}\Big ]$,

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} \left| \int _r^t \nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _sds \right|^{2q}\right] }\\&\quad \le {\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} { \left( \int _r^t \left|\nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( Y^{\epsilon ,w^\epsilon }_s\right) \right|^2ds\right) ^{q} \left( \int _0^1\left|{\dot{u}}^\epsilon _s \right|^2 ds\right) ^{q}}\right] }\\&\quad \le C\rho ^{q-1}{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}} { \int _r^t {1+\left|Y^{\epsilon ,w^\epsilon }_s \right|^{2q(D_f+D_g)}} ds}\right] }\le C\rho ^{q-1}, \end{aligned}$$

where the last inequality is once again obtained using Lemma 11. $\square$

Lemma 13

Assume $w^\epsilon \in {\mathcal {S}}$ is a control such that

$$\begin{aligned} \sup _{\epsilon>0}\left\Vert w^\epsilon \right\Vert ^2_{{\mathcal {S}}}=\sup _{\epsilon >0}\int _0^1 \left[ \left|{\hat{u}}^\epsilon _s \right|^2+\left|{\hat{v}}^\epsilon _s \right|^2 \right] ds< N \end{aligned}$$

for some finite constant N. Under Condition H1, for $0<\alpha \le 1/2$, we have the almost sure Hölder estimate

$$\begin{aligned} {\left|Y^{\epsilon ,w^\epsilon } \right|_{\alpha }}\le \frac{C}{\sqrt{\epsilon }}. \end{aligned}$$

Proof

Without loss of generality, let us assume $t>r$. The dissipative property of the drift coefficient of $Y^{\epsilon ,w^\epsilon }$ and Itô’s formula yield

$$\begin{aligned} Y^{\epsilon ,w^\epsilon }_t&=e^{-\frac{1}{\epsilon }\Gamma (t-r)}Y^{\epsilon ,w^\epsilon }_r+\int _r^t \frac{1}{\epsilon } e^{-\frac{1}{\epsilon }(t-s)}\zeta (Y^{\epsilon ,w^\epsilon })ds+\int _r^t\frac{h(\epsilon )}{\sqrt{\epsilon }}e^{-\frac{1}{\epsilon }(t-s)}\sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{v}}^\epsilon _s ds\\&\quad +\,\int _r^t\frac{1}{\sqrt{\epsilon }}e^{-\frac{1}{\epsilon }(t-s)}\sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) dB_s. \end{aligned}$$

Now, by subtracting $Y^{\epsilon ,w^\epsilon }_r$ from both sides and applying Hölder’s inequality along with the Burkhölder–Davis–Gundy inequality, we get

$$\begin{aligned}&{\mathbb {E}}{\left[ \left|Y^{\epsilon ,w^\epsilon }_t-Y^{\epsilon ,w^\epsilon }_r \right|\right] }\nonumber \\ {}&\quad \le \left|e^{-\frac{1}{\epsilon }\Gamma (t-r)}-1 \right| {\mathbb {E}}{\left[ \left|Y^{\epsilon ,w^\epsilon }_r \right|\right] }+\frac{1}{\epsilon }\sqrt{\int _r^t e^{-\frac{2}{\epsilon }\Gamma (t-s)} ds}{\mathbb {E}}{\left[ \sqrt{\int _0^1 \left|\zeta (Y^{\epsilon ,w^\epsilon }) \right|^2 ds }\right] }\nonumber \\&\qquad + \,\frac{h(\epsilon )}{\sqrt{\epsilon }}\sqrt{\int _r^t e^{-\frac{2}{\epsilon }\Gamma (t-s)} ds}{\mathbb {E}}{\left[ \sqrt{\int _0^1 \left|{\dot{v}}^\epsilon _s \right|^2 ds }\right] }\nonumber \\&\qquad +\,\frac{1}{\sqrt{\epsilon }}{\mathbb {E}}{\left[ \sqrt{\int _r^t e^{-\frac{2}{\epsilon }\Gamma (t-s)}\left|\sigma (Y^{\epsilon ,w^\epsilon }_s)\sigma ^\top (Y^{\epsilon ,w^\epsilon }_s) \right|ds}\right] }. \end{aligned}$$

(45)

To bound the first term on the right-hand side, we combine the second estimate in Lemma 11 and the fact that $e^{-\frac{1}{\epsilon }\Gamma (t-r)}-1=\frac{1}{\epsilon }\int _r^t e^{-\frac{1}{\epsilon }\Gamma (t-s)}ds\le \left|t-r \right|$. For the second term, note that $\int _r^t e^{-\frac{2}{\epsilon }\Gamma (t-s)}ds=C\epsilon \left|t-r \right|.$ Moreover, the sublinearity of $\zeta (y)$ and the first estimate in Lemma 11 yield a finite bound on the expression ${\mathbb {E}}{\left[ \sqrt{\int _0^1 \left|\zeta (Y^{\epsilon ,w^\epsilon }) \right|^2 ds }\right] }.$ The third term on the right-hand side of (45) can be treated similarly with the help of Lemma 10. Regarding the last term, recall that $\sigma (y)\sigma ^T(y)$ is bounded in Condition H1. Thus, we have

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le r,t\le 1} \left|Y^{\epsilon ,w^\epsilon }_t-Y^{\epsilon ,w^\epsilon }_r \right|\right] }\le C\frac{1}{\sqrt{\epsilon }}\left|t-r \right|^{1/2}. \end{aligned}$$

The Kolmogorov continuity theorem then yields the almost sure Hölder continuity of $Y^{\epsilon ,w^\epsilon }$. $\square$

Lemma 14

Assume $w^\epsilon \in {\mathcal {S}}$ is a control such that

$$\begin{aligned} \sup _{\epsilon>0}\left\Vert w^\epsilon \right\Vert ^2_{{\mathcal {S}}}=\sup _{\epsilon >0}\int _0^1 \left[ \left|{\hat{u}}^\epsilon _s \right|^2+\left|{\hat{v}}^\epsilon _s \right|^2 \right] ds< N \end{aligned}$$

for some finite constant N. Under Conditions H1 and H2-A or H2-B, there exist a constant C and $\epsilon _0$ small enough such that for $0<\beta \le 1/2$,

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{\epsilon<\epsilon _0}\left|X^{\epsilon ,w^\epsilon } \right|_{\beta }\right] }<C. \end{aligned}$$

Proof

We begin by proving the result under Conditions H1 and H2-A. According to Condition H2-A, f(x, y) is Lipschitz-continuous and bounded, so that f(x, y) is also $\gamma$-Hölder continuous for $0<\gamma \le 1$. This further implies

$$\begin{aligned}&{\mathbb {E}}{\left[ \left|f\left( X^{\epsilon ,w^\epsilon }_t,Y^{\epsilon ,w^\epsilon }_t\right) -f\left( X^{\epsilon ,w^\epsilon }_r,Y^{\epsilon ,w^\epsilon }_r\right) \right|\right] } \\ {}&\quad \le {\mathbb {E}}{\left[ \left|f\left( X^{\epsilon ,w^\epsilon }_t,Y^{\epsilon ,w^\epsilon }_t\right) -f\left( X^{\epsilon ,w^\epsilon }_r,Y^{\epsilon ,w^\epsilon }_t\right) \right|\right] }\\&\qquad +\,{\mathbb {E}}{\left[ \left|f\left( X^{\epsilon ,w^\epsilon }_r,Y^{\epsilon ,w^\epsilon }_t\right) -\left( X^{\epsilon ,w^\epsilon }_r,Y^{\epsilon ,w^\epsilon }_r\right) \right|\right] }\\&\quad \le C{\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon }_t-X^{\epsilon ,w^\epsilon }_r \right|\right] }+{\mathbb {E}}{\left[ \left|Y^{\epsilon ,w^\epsilon }_t-Y^{\epsilon ,w^\epsilon }_r \right|^{\gamma }\right] }\\&\quad \le C\left( {\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon }_t-X^{\epsilon ,w^\epsilon }_r \right|\right] }+\epsilon ^{-\frac{\gamma }{2}}\left|t-r \right|^\gamma \right) , \end{aligned}$$

and hence that for $0<\gamma \le 1$,

$$\begin{aligned} {\mathbb {E}}{\left[ \left|f\left( X^{\epsilon ,w^\epsilon },Y^{\epsilon ,w^\epsilon }\right) \right|_\gamma \right] }\le C\left( {\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon } \right|_\gamma \right] }+\epsilon ^{-\frac{\gamma }{2}}\right) . \end{aligned}$$

This last estimate, together with the Young–Loéve inequality in Lemma 7, implies that for $1-H<\beta \le 1$,

$$\begin{aligned} {\mathbb {E}}{\left[ \left|\int _r^t f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) dW^H_s \right|\right] }&\le C{\mathbb {E}}{\left[ \left|f\left( X^{\epsilon ,w^\epsilon },Y^{\epsilon ,w^\epsilon }\right) \right|_{\beta } \right] }\left|t-r \right|^H\nonumber \\&\le C{\mathbb {E}}{\left[ {\left|X^{\epsilon ,w^\epsilon } \right|_\beta +\epsilon ^{-\frac{\beta }{2}}}\right] }\left|t-r \right|^H. \end{aligned}$$

(46)

Meanwhile, a similar estimate to the one stated in part (i) of Lemma 12 states that

$$\begin{aligned} {\mathbb {E}}{\left[ \left|\int _0^t f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _sds \right|\right] }&\le C\left|r-t \right|^{\frac{1}{2}}. \end{aligned}$$

Moreover, boundedness of g(x, y) in Condition H2-A yields

$$\begin{aligned} {\mathbb {E}}{\left[ \left|\int _r^t g\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) ds \right| \right] }\le C\left|t-r \right|. \end{aligned}$$

Thus, using the estimate ${\mathbb {E}}{\left[ \left|Y^{\epsilon ,w^\epsilon } \right|_{\alpha }\right] }\le C\frac{1}{\sqrt{\epsilon }},\alpha \le \frac{1}{2}$ in Lemma 13 (which requires Condition H1), we can deduce that, for $1-H<\beta \le 1/2$,

$$\begin{aligned} {\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon }_t-X^{\epsilon ,w^\epsilon }_r \right|\right] }\le C\left( {\mathbb {E}}{\left[ \sqrt{\epsilon }\left|X^{\epsilon ,w^\epsilon } \right|_\beta \right] } \left|t-r \right|^H+\left|t-r \right|^{\frac{1}{2}}+\left|t-r \right|\right) , \end{aligned}$$

and consequently,

$$\begin{aligned} {\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon } \right|_\beta \right] }\le C\left( {\mathbb {E}}{\left[ \sqrt{\epsilon }\left|X^{\epsilon ,w^\epsilon } \right|_\beta \right] } \left|t-r \right|^{H-\beta }+\left|t-r \right|^{\frac{1}{2}-\beta }+\left|t-r \right|^{1-\beta }\right) . \end{aligned}$$

Now, by choosing $\epsilon _0$ small enough, we get ${\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon } \right|_\beta \right] }\le C$ for some constant C. Since for $0<\beta _1\le \beta _2\le 1$, $\beta _2$-Hölder continuity of $X^{\epsilon ,w^\epsilon }$ implies $\beta _1$-Hölder continuity, the conclusion follows.

We now present a proof of the claim under Conditions H1 and H2-B. Under Condition H2-B, f(y) is $M_f$-Hölder continuous, while $Y^{\epsilon ,w^\epsilon }$ is $\frac{1}{2}$-Hölder continuous by Lemma 13, so that

$$\begin{aligned} {\mathbb {E}}{\left[ \left|f\left( Y^{\epsilon ,w^\epsilon }_t\right) -f\left( Y^{\epsilon ,w^\epsilon }_r\right) \right|\right] } \le C\left|Y^{\epsilon ,w^\epsilon }_t-Y^{\epsilon ,w^\epsilon }_r \right|^{M_f} \le C\epsilon ^{-\frac{M_f}{2}}\left|t-r \right|^{\frac{M_f}{2}} \end{aligned}$$

or equivalently

$$\begin{aligned} {\mathbb {E}}{\left[ \left|f\left( Y^{\epsilon ,w^\epsilon }\right) \right|_{\frac{M_f}{2}}\right] } \le C\epsilon ^{-\frac{M_f}{2}}. \end{aligned}$$

Then, the Young–Loéve inequality in Lemma 7 implies that, for $1-\frac{M_f}{2}<K\le H$,

$$\begin{aligned}&{\mathbb {E}}{\left[ \left|\int _r^t f\left( Y^{\epsilon ,w^\epsilon }_s\right) dW^H_s \right|\right] }\nonumber \\&\quad \le {{\left|Y^{\epsilon ,w^\epsilon } \right|}_{\frac{M_f}{2}}}\left|t-r \right|^{\frac{M_f}{2}+K}\left|W^H \right|_K+\left|f(Y^{\epsilon ,w^\epsilon }_r) \right|\left|W^H_t-W^H_r \right|\nonumber \\&\quad \le C\left( {\left|Y^{\epsilon ,w^\epsilon } \right|}_{\frac{M_f}{2}}\left|t-r \right|^{\frac{M_f}{2}+K}+{\mathbb {E}}{\left[ \sup _{t\in [0,1]} \left|Y^{\epsilon ,w^\epsilon }_t \right|^{D_f}\right] } \left|t-r \right|^K\right) \nonumber \\&\quad \le C\left( \epsilon ^{-\frac{M_f}{2}}+\epsilon ^{-\frac{1}{2}}\right) \left|t-r \right|^K \le C\epsilon ^{-\frac{1}{2}}\left|t-r \right|^K, \end{aligned}$$

(47)

where the first inequality is obtained by Condition H2-B and the last inequality is a consequence of the estimate ${\mathbb {E}}{\left[ \left|Y^{\epsilon ,w^\epsilon } \right|_{\alpha }\right] }\le C\frac{1}{\sqrt{\epsilon }},\alpha \le \frac{1}{2}$ in Lemma 13 (which requires Condition H1). Moreover, similar calculations to those performed in the proof of part (ii) of Lemma 12 yield that, for any q in $\bigg (1,\frac{1}{D_f}\bigg ]$,

$$\begin{aligned} {\mathbb {E}}{\left[ \left|\int _r^t f\left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _sds \right|\right] }&\le C\left|t-r \right|^{\frac{1}{2}-\frac{1}{2q}}, \end{aligned}$$

as well as

$$\begin{aligned} {\mathbb {E}}{\left[ \left|\int _r^t g\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) ds \right| \right] }\le C\left|t-r \right|^{\frac{1}{2}}. \end{aligned}$$

Consequently, we have

$$\begin{aligned} {\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon }_t-X^{\epsilon ,w^\epsilon }_r \right|\right] }\le C\left( \left|t-r \right|^K+\sqrt{\epsilon }h(\epsilon )\left|t-r \right|^{\frac{1}{2}-\frac{1}{2q}}+\left|t-r \right|^{\frac{1}{2}}\right) . \end{aligned}$$

By choosing $\epsilon _0$ small enough and noting that $K>1-\frac{M_f}{2}\ge \frac{1}{2}$, we arrive at

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{\epsilon<\epsilon _0}\left|X^{\epsilon ,w^\epsilon } \right|_\beta \right] }<C \end{aligned}$$

for $0\le \beta \le \frac{1}{2}$. $\square$

Lemma 15

Assume $w^\epsilon \in {\mathcal {S}}$ is a control such that

$$\begin{aligned} \sup _{\epsilon>0}\left\Vert w^\epsilon \right\Vert ^2_{{\mathcal {S}}}=\sup _{\epsilon >0}\int _0^1 \left[ \left|{\hat{u}}^\epsilon _s \right|^2+\left|{\hat{v}}^\epsilon _s \right|^2 \right] ds< N \end{aligned}$$

for some finite constant N. Then, the following two assertions hold.

• (i) Under Conditions H1 and H2-A, there exists a constant C such that for any $\beta$ in $(1-H,1]$,

  $$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\int _0^t f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) dW^H_s \right|\right] } \le C{\epsilon ^{-\frac{\beta }{2}}} \end{aligned}$$

  and

  $$\begin{aligned} {\mathbb {E}}{\left[ \left|\sup _{t\in [0,1]} \int _0^t\nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) d{W}^H_s \right|\right] }&\le C\epsilon ^{-\frac{1}{2}}. \end{aligned}$$

• (ii) Under Conditions H1 and H2-B, there exists a constant C such that

  $$\begin{aligned} {\mathbb {E}}{\left[ \sup _{t\in [0,1]}\left|\int _0^t f\left( Y^{\epsilon ,w^\epsilon }_s\right) dW^H_s \right|\right] } \le C{\epsilon ^{-\frac{M_f}{2}}} \end{aligned}$$

  and

  $$\begin{aligned} {\mathbb {E}}{\left[ \sup _{t\in [0,1]}\left|\int _0^t \nabla _x\phi (X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s)f(Y^{\epsilon ,w^\epsilon }_s)dW^H_s \right|\right] }\le C\epsilon ^{-\frac{M_k}{2}}. \end{aligned}$$

Proof

We begin by proving part (i). The first estimate is immediate based on Lemma 14 and the estimate at (46). Regarding the second estimate, the inequality at (43) and Condition H2-A imply that $\nabla _x\phi (x,y)f(x,y)$ is Lipschitz continuous. Hence, by the Young–Loéve inequality for $1-H<\beta \le 1$, we have

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\int _0^t\nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) d{W}^H_s \right|\right] }\\&\quad \le {\mathbb {E}}{\left[ \left|\nabla _x\phi \left( X^{\epsilon ,w^\epsilon },Y^{\epsilon ,w^\epsilon }\right) f(X^{\epsilon ,w^\epsilon },Y^{\epsilon ,w^\epsilon }) \right|_\beta \right] }\\&\quad \le \left|\nabla _x\phi (x,y)f(x,y) \right|_{{\text {Lip}}}{\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon } \right|_\beta +\left|Y^{\epsilon ,w^\epsilon } \right|_\beta \right] }\\&\quad \le C\left( 1+\epsilon ^{-\frac{1}{2}}\right) , \end{aligned}$$

where the last inequality is a consequence of Lemmas 13 and 14.

We now proceed to the proof of part (ii). For the first estimate, we perform a similar calculation to the one that was done at (47) (this requires Conditions H1 and H2-B) and get

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{t\in [0,1]}\left|\int _0^t f\left( Y^{\epsilon ,w^\epsilon }_s\right) dW^H_s \right|\right] } \le C\left( {\left|Y^{\epsilon ,w^\epsilon } \right|}_{\frac{M_f}{2}}+{\mathbb {E}}{\left[ (y_0)^{D_f}\right] } \right) \le C{\epsilon ^{-\frac{M_f}{2}}}. \end{aligned}$$

Next, under Conditions H1 and H2-B, the $M_k$-Hölder continuity of $\nabla _x\phi (x,y)f(x)$ together with the estimates in Lemmas 13 and 14 yields

$$\begin{aligned}&{\mathbb {E}}{\left[ \left|\nabla _x\phi (X^{\epsilon ,w^\epsilon }_r,Y^{\epsilon ,w^\epsilon }_r)f(Y^{\epsilon ,w^\epsilon }_r)-\nabla _x\phi (X^{\epsilon ,w^\epsilon }_t,Y^{\epsilon ,w^\epsilon }_t)f(Y^{\epsilon ,w^\epsilon }_t) \right|\right] }\\&\quad \le {\mathbb {E}}{\left[ \left|X^{\epsilon ,w^\epsilon }_r-X^{\epsilon ,w^\epsilon }_t \right|^{M_k}\right] }+{\mathbb {E}}{\left[ \left|Y^{\epsilon ,w^\epsilon }_r-Y^{\epsilon ,w^\epsilon }_t \right|^{M_k}\right] }\\&\quad \le C\left( 1+\epsilon ^{-\frac{M_k}{2}}\right) \left|r-t \right|^{\frac{M_k}{2}}, \end{aligned}$$

so that

$$\begin{aligned} {\mathbb {E}}{\left[ \left|\nabla _x\phi (X^{\epsilon ,w^\epsilon },Y^{\epsilon ,w^\epsilon })f(Y^{\epsilon ,w^\epsilon }) \right|_{\frac{M_k}{2}}\right] }\le C\epsilon ^{-\frac{M_k}{2}}. \end{aligned}$$

Therefore, as $\frac{M_k}{2}+H>1$ in Condition H2-B, we can apply the Young–Loéve inequality to obtain

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{t\in [0,1]}\left|\int _0^t \nabla _x\phi (X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s)f(Y^{\epsilon ,w^\epsilon }_s)dW^H_s \right|\right] }\\ {}&\quad \le C\bigg ( \left|W^H \right|_H {\mathbb {E}}{\left[ \left|\nabla _x\phi (X^{\epsilon ,w^\epsilon },Y^{\epsilon ,w^\epsilon })f(Y^{\epsilon ,w^\epsilon }) \right|_{\frac{M_k}{2}}\right] }\\ {}&\quad +{{\mathbb {E}}{\left[ \left|\nabla _x\phi (x_0,y_0)f(y_0) \right|\right] }}\bigg )\\&\quad \le C\epsilon ^{-\frac{M_k}{2}}. \square \end{aligned}$$

Lemma 16

Assume $w^\epsilon \in {\mathcal {S}}$ is a control such that

$$\begin{aligned} \sup _{\epsilon>0}\left\Vert w^\epsilon \right\Vert ^2_{{\mathcal {S}}}=\sup _{\epsilon >0}\int _0^1 \left[ \left|{\hat{u}}^\epsilon _s \right|^2+\left|{\hat{v}}^\epsilon _s \right|^2 \right] ds< N \end{aligned}$$

for some finite constant N. Under Conditions H1 and either H2-A or H2-B, there exists a constant C such that

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1} \left|\eta ^{\epsilon ,w^\epsilon }_t \right|^2\right] } < C. \end{aligned}$$

Furthermore, this implies for any $\rho >0$,

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}}\left|\int _{r}^{t} \frac{1}{\sqrt{\epsilon }h(\epsilon )}\left( {\bar{g}}(X^{\epsilon ,w^\epsilon }_s)-{\bar{g}}({\bar{X}}_s)\right) ds \right|\right] }\le C\rho . \end{aligned}$$

Proof

Under Condition H2-A or H2-B, $\nabla _x{\bar{g}}(x)$ is bounded. This fact, combined with equation (27) and the fact that $X^{\epsilon ,w^\epsilon }$ converges to ${\bar{X}}$ in probability, implies that there exists some constant C such that

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\int _{0}^{t} \frac{1}{\sqrt{\epsilon }h(\epsilon )}\left( {\bar{g}}(X^{\epsilon ,w^\epsilon }_s)-{\bar{g}}({\bar{X}}_s)\right) ds \right|^2\right] }\le C\int _0^1 {\mathbb {E}}{\left[ \sup _{0\le r\le s}\left|\eta ^{\epsilon ,w^\epsilon }_r \right|^2\right] }ds. \end{aligned}$$

(48)

In addition, based on equation (25), we have

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\eta ^{\epsilon ,w^\epsilon }_t \right|^2\right] }&\le C\bigg ({\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\int _0^t \nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) {\dot{v}}^\epsilon _sds \right|^2\right] }\nonumber \\&\quad +\, {\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\int _{0}^{t} \frac{1}{\sqrt{\epsilon }h(\epsilon )}\left( {\bar{g}}(X^{\epsilon ,w^\epsilon }_s)-{\bar{g}}({\bar{X}}_s)\right) ds \right|^2\right] }\nonumber \\&\quad +\,{\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\frac{1}{h(\epsilon )}\int _0^tf\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) d{W}^H_s \right|^2\right] }\nonumber \\&\quad +\,{\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\int _0^tf\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _sds \right|^2\right] } +{\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|R^\epsilon _2(t) \right|^2\right] }\bigg ), \end{aligned}$$

(49)

with

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|R^\epsilon _2(t) \right|^2\right] }&\le C\bigg ({\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\frac{\sqrt{\epsilon }}{h(\epsilon )}\left( \phi \left( X^{\epsilon ,w^\epsilon }_t,Y^{\epsilon ,w^\epsilon }_t\right) -\phi (x_0,y_0)\right) \right|^2\right] }\\&\quad +\,{\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\frac{\sqrt{\epsilon }}{h(\epsilon )}\int _0^t \nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) g\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) ds \right|^2\right] }\\&\quad +\,{\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\epsilon \int _0^t\nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) {\dot{u}}^\epsilon _s ds \right|^2\right] }\\&\quad +\,{\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\frac{{\epsilon }}{h(\epsilon )}\int _0^t\nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) d{W}^H_s \right|^2\right] }\\&\quad +\,{\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\frac{1}{h(\epsilon )}\int _0^t\nabla _y\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) \sigma \left( Y^{\epsilon ,w^\epsilon }_s\right) dB_s \right|^2\right] }\bigg ). \end{aligned}$$

We will estimate the terms on the right-hand side of (49), starting with those which contain Young’s integrals. Condition H2-A guarantees that there exists some $\beta$ in [0, 1] such that $\beta +H>1$ and $h(\epsilon )^{-1}\epsilon ^{-\frac{\beta }{2}}\rightarrow 0$ as $\epsilon \rightarrow 0$, so that part (i) of Lemma 15 (which requires Conditions H1 and H2-A) yields

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\frac{1}{h(\epsilon )}\int _0^tf\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) d{W}^H_s \right|^2\right] }\le Ch(\epsilon )^{-2}\epsilon ^{-\beta }\rightarrow 0. \end{aligned}$$

Part (i) of Lemma 15 also implies that

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{t\in [0,1]} \left|\frac{{\epsilon }}{h(\epsilon )}\int _0^t\nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) d{W}^H_s \right|^2\right] }\\ {}&\qquad \le C\frac{1}{h(\epsilon )^2}\rightarrow 0. \end{aligned}$$

Meanwhile, under Condition H2-B, we use part (ii) of Lemma 15 to get, as $\epsilon \rightarrow 0$,

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\frac{1}{h(\epsilon )}\int _0^tf\left( Y^{\epsilon ,w^\epsilon }_s\right) d{W}^H_s \right|^2\right] }\le Ch(\epsilon )^{-2}\epsilon ^{-M_f}\rightarrow 0 \end{aligned}$$

and

$$\begin{aligned}&{\mathbb {E}}{\left[ \sup _{t\in [0,1]} \left|\frac{{\epsilon }}{h(\epsilon )}\int _0^t\nabla _x\phi \left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) f\left( X^{\epsilon ,w^\epsilon }_s,Y^{\epsilon ,w^\epsilon }_s\right) d{W}^H_s \right|^2\right] }\\ {}&\qquad \le Ch(\epsilon )^{-2}\epsilon ^{2-M_k}\rightarrow 0. \end{aligned}$$

The remaining terms on the right-hand side of (49), except the term

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1}\left|\int _{0}^{t} \frac{1}{\sqrt{\epsilon }h(\epsilon )}\left( {\bar{g}}(X^{\epsilon ,w^\epsilon }_s)-{\bar{g}}({\bar{X}}_s)\right) ds \right|^2\right] }, \end{aligned}$$

are bounded by using Lemmas 11 and 12 (which require Conditions H1 and H2-B). Thus, it follows from the estimates at (48) and (49) that

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1} \left|\eta ^{\epsilon ,w^\epsilon }_t \right|^2\right] }\le C_1+C_2\int _0^1{\mathbb {E}}{\left[ \sup _{0\le r\le s} \left|\eta ^{\epsilon ,w^\epsilon }_r \right|^2\right] }ds. \end{aligned}$$

An application of Gronwall’s inequality then yields the first claim of our statement, which is

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le t\le 1} \left|\eta ^{\epsilon ,w^\epsilon }_t \right|^2\right] }\le C. \end{aligned}$$

For the second claim, we proceed similarly to the derivation of the estimate at (48). Then, for $\rho >0$,

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}}\left|\int _{r}^{t} \frac{1}{\sqrt{\epsilon }h(\epsilon )}\left( {\bar{g}}(X^{\epsilon ,w^\epsilon }_s)-{\bar{g}}({\bar{X}}_s)\right) ds \right|\right] }&\le C{\mathbb {E}}{\left[ \sup _{\begin{array}{c} 0\le r,t\le 1 \\ \left|r-t \right|<\rho \end{array}}\int _r^t {\left|\eta ^{\epsilon ,w^\epsilon }_s \right|}ds\right] }\\&\le C\rho {\mathbb {E}}{\left[ \sup _{0\le t\le 1} \left|\eta ^{\epsilon ,w^\epsilon }_t \right|^2\right] }\le C\rho . \end{aligned}$$

$\square$

Lemma 17

Let $R_2^\epsilon$ be the remainder term that appears in equation (27). Under Conditions H1 and either H2-A or H2-B , it holds that $R_2^\epsilon \rightarrow 0$ in $C([0,1];{\mathbb {R}}^{n})$ in probability along a subsequence.

Proof

For the purpose of identifying the limit of $R_2^\epsilon$, we invoke the Skorokhod representation theorem and assume that $X^{\epsilon ,w^\epsilon }\rightarrow {\bar{X}}$ a.s. in $C([0,1];{\mathbb {R}}^{n})$ as $\epsilon \rightarrow 0$. As ${\bar{X}}$ is bounded under Condition H2-A or H2-B, the dominated convergence theorem implies that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}{\mathbb {E}}{\left[ \sup _{0\le s\le 1}\left|X^{\epsilon ,w^\epsilon }_s- {\bar{X}}_s \right|^2\right] }=0. \end{aligned}$$

(50)

Now, we employ the bound (28) and get

$$\begin{aligned} {\mathbb {E}}{\left[ \sup _{0\le s\le 1}\left|R^\epsilon _2(s) \right|\right] }&\le {\mathbb {E}}{\left[ \int _0^1 \left|\nabla ^2_x {\bar{g}} \right|_\infty \left|\eta ^{\epsilon ,w^\epsilon }_s \right| \left|X^{\epsilon ,w^\epsilon }_s-{\bar{X}}_s \right|ds\right] }\\&\le C{\mathbb {E}}{\left[ \sup _{0\le s\le 1} \left|\eta ^{\epsilon ,w^\epsilon }_s \right| \sup _{0\le s\le 1} \left|X^{\epsilon ,w^\epsilon }_s-{\bar{X}}_s \right|\right] }\\&\le C{\mathbb {E}}{\left[ \sup _{0\le s\le 1} \left|\eta ^{\epsilon ,w^\epsilon }_s \right|^2\right] }{\mathbb {E}}{\left[ \sup _{0\le s\le 1} \left|X^{\epsilon ,w^\epsilon }_s-{\bar{X}}_s \right|^2\right] }\\&\le C{\mathbb {E}}{\left[ \sup _{0\le s\le 1} \left|X^{\epsilon ,w^\epsilon }_s-{\bar{X}}_s \right|^2\right] }, \end{aligned}$$

In particular, the second inequality is due to the boundedness of $\nabla ^2_x{\bar{g}}$ implied by either Condition H2-A or H2-B. The last inequality is a consequence of Lemma 16 (which requires Conditions H1 and either H2-A or H2-B). (50) then gives us the desired limit. $\square$