A Milstein-type method for highly non-linear non-autonomous time-changed stochastic differential equations

Time-changed stochastic processes and time-changed stochastic differential equations (SDEs) have been attracting increasing attentions in the past decades, as they are one of the important tools to describe sub-diffusion processes and their close relation with deterministic fractional differential equations (DFDE) [31].

In [25], Meerschaert and Scheffler gave a detailed discussion on the time process used for changing times and established a fundamental limit theorem that links some continuous-time random walks with infinite mean waiting times with a class of time-changed stochastic processes. Some important properties and essential inequalities of time-changed fractional Brownian motion were obtained by Deng and Schilling in [4].

The existence and uniqueness theorem for time-changed SDEs and many useful tools were obtained by Kobayashi in [16]. Stabilities in different senses of all kinds of stochastic equations were broadly discussed: Wu in [34] investigated SDEs driven time-changed Brownian motion; Nane and Li in [27, 28] studied the case when the driven noise is the time-changed Lévy noise; Zhang and Yuan focused on time-changed stochastic functional differential equations in [36]; Yin et al. considered the impulsive effects on stabilities in [35] for a class of time-changed SDEs; Shen et al. in [30] discussed distribution dependent SDEs driven by time-changed Brownian motions. Li et al. discussed some theoretical results of the time-changed McKean-Vlasov SDE in [20], which is also a distribution dependent SDE.

Time-changed processes and time-changed SDEs are widely applied in modelling financial markets. Magdziarz introduced sub-diffusive Black-Scholes formula by using the classical geometric Brownian motion with the inverse $\alpha$-stable subordinator [22]. Magdziarz et al. proposed the sub-diffusive version of the Bachelier Model and investigated its application in the option pricing [24]. Janczura et al. studied the time-changed Ornstein-Uhlenbeck process that is driven by the $\alpha$-stable process and fitted the data from emerging markets in this model [12]. For connections between time-changed processes and various DFDE, we refer the readers to [2, 9, 23, 26] and references therein.

Since the following two main reasons, numerical approximations to time-changed SDEs become essential. (1) Explicit forms of true solutions to time-changed SDEs are hardly found. (2) Applications of time-changed SDE models in practice often require a considerable number of sample paths to conduct statistical learnings like estimations, tests and predictions based on observed data. In this case, even explicit expressions of true solutions to some types of time-changed SDE models are available, performing those calculations without the aid of computer simulations is highly unlikely.

When transition probabilities of solutions to time-changed SDEs are needed to be simulated, the typical approaches used are discretising the corresponding deterministic fractional differential equations. There are fruitful works on numerical methods for DFDE and a far-from-complete list of them includes [5, 6, 17, 32] among many others.

In this paper, we focus on another important aspect of numerical approximations to time-changed SDEs, i.e. numerical simulations of sample paths of solutions.

In this aspect, Kobayashi and collaborators studied different numerical methods for time-changed SDEs with different structures, when the global Lipschitz conditions are imposed on the spatial variables in the coefficients. The convergences in both the strong and weak senses of the Euler–Maruyama (EM) method for a class of time-changed SDEs were proved by Jum and Kobayashi in [15], which, to our best knowledge, is the first work to study simulations of sample paths of solutions to time-changed SDEs. More recently, Jin and Kobayashi investigated some Euler-type and Milstein-type methods for more general type of time-changed SDEs in [13, 14]. One of the main differences in terms of techniques used between [15] and [13, 14] is that the duality principle established in [16] was employed in [15] but not in [13, 14]. Briefly speaking, the duality principle reveals the relation between the classical SDEs and time-changed counterpart, which enables numerical methods for time-changed SDEs to be constructed by using numerical methods for classical SDEs directly. For time-changed McKean-Vlasov SDEs, Wen et al. considered the numerical method in [33].

In the case that some super-linear terms are allowed to appear in the coefficients, implicit methods and modified explicit methods are usually good alternatives as the classical Euler-type and Milstein-type methods may not be convergent [11]. When some super-linear growth conditions are imposed on the spatial variables in the drift coefficient of time-changed SDEs, Deng and Liu studied the semi-implicit EM method in [3], Liu et al. investigated the truncated EM method in [21] with the help of the duality principle, while Li et al. in [18] also discussed the truncated-type Euler method but without employing the duality principle.

In this paper, we also focus on numerical methods for time-changed SDEs with super-linear coefficients. Compared with [3, 21, 18], we consider the numerical method with the higher convergence order by proposing a Milstein-type method with the truncating techniques to suppress super-linear terms. Due to the higher convergence order, compared with those Euler-type methods Milstein-type methods are more suitable for the multi-level Monte Carlo that is quite popular for applications in finance [7, 8].

Let $(\Omega_{W},{\cal F}^{W},\mathbb{P}_{W})$ be a complete probability space with a filtration $\{{\cal F}^{W}_{t}\}_{t\geq 0}$ being right continuous and increasing, while ${\cal F}^{W}_{0}$ contains all $\mathbb{P}_{W}$-null sets. Let $W(t)$ be a one-dimensional Wiener process defined in that probability space and is ${\cal F}^{W}_{t}$-adapted. $\mathbb{E}_{W}$ denotes the expectation with respect to $\mathbb{P}_{W}$.

Let $(\Omega_{D},{\cal F}^{D},\mathbb{P}_{D})$ be another complete probability space with a filtration $\left\{{\cal F}^{D}_{t}\right\}_{t\geq 0}$. $D(t)$ denotes a one-dimensional ${\cal F}^{D}_{t}$-adapted strictly increasing Lévy process on $[0,\infty)$ starting from $D(0)=0$ defined on $(\Omega_{D},{\cal F}^{D},\mathbb{P}_{D})$. Let $\mathbb{E}_{D}$ denote the expectation with respect to $\mathbb{P}_{D}$. For detailed introductions and discussions on such a $D(t)$, we refer the readers to [1, 29]

In this paper, $W(t)$ and $D(t)$ are assumed to be independent. Define the product probability space by $(\Omega,{\cal F},\mathbb{P}):=(\Omega_{W}\times\Omega_{D},{\cal F}^{W}\otimes{\cal F}^{D},\mathbb{P}_{W}\otimes\mathbb{P}_{D})$. Let $\mathbb{E}$ denote the expectation under the probability measure $\mathbb{P}$. It is clear that $\mathbb{E}(\cdot)=\mathbb{E}_{D}\left(\mathbb{E}_{W}(\cdot)\right)=\mathbb{E}_{W}\left(\mathbb{E}_{D}(\cdot)\right)$.

For $x\in\mathbb{R}^{d}$, $|x|$ denotes the Euclidean norm. The transposition of $x$ is denoted by $x^{\mathrm{T}}$. For two real numbers $a$ and $b$, set $a\vee b=\max(a,b)$ and $a\wedge b=\min(a,b)$. For a given set $G$, its indicator function is denoted by $\mathbf{1}_{G}$.

Since $D(t)$ is strictly increasing, we define the inverse of $D(t)$ by

$\displaystyle E(t):=\inf\{s\geq 0\,;\,D(s)>t\},\quad t\geq 0.$

Then, the $E(t)$ is used for changing time, as $t\mapsto E(t)$ is continuous and non-decreasing. The process $W(E(t))$ is called a time-changed Wiener process and $W(E(t))$ is regarded as a sub-diffusive process. For the simplicity of notations, we consider the one-dimensional $W(E(t))$ in our work. When $W(t)$ is a multi-dimensional Wiener process and the same $E(t)$ is used for changing time in each entry of $W(t)$, the results in this paper should still hold. But if different $E(t)$s are used to change times in different entries of $W(t)$, our results may not be applicable.

The time-changed SDEs considered in this paper take the following form, For any $T>0$ and $t\in[0,T]$

$\displaystyle dY(t)=f(t,Y(t))dE(t)+g(t,Y(t))dW(E(t)),~{}~{}~{}Y(0)=Y_{0},$

(1)

with $\mathbb{E}|Y_{0}|^{q}<\infty$ for all $q>0$, where $f:\mathbb{R}_{+}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and $g:\mathbb{R}_{+}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$.

Before we impose assumptions on the coefficients of (1), we present some tedious but helpful notations. For any $y=(y^{1},y^{2},...,y^{d})\in\mathbb{R}^{d}$ and and any $t\in[0,T]$, define

$\displaystyle Lg(t,y)=\sum_{l=1}^{d}g^{l}(t,y)G^{l}(t,y),$

where $g=(g^{1},g^{2},...,g^{d})^{T}$, $g^{l}:\mathbb{R}_{+}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ and

$\displaystyle G^{l}(t,y)=\left(\dfrac{\partial g^{1}(t,y)}{\partial y^{l}},\dfrac{\partial g^{2}(t,y)}{\partial y^{l}},...,\dfrac{\partial g^{d}(t,y)}{\partial y^{l}}\right)^{\mathrm{T}}.$

It can be observed from Assumption 1 that for all $t\in[0,T]$ and any $x\in\mathbb{R}^{d}$

$\displaystyle|f(t,x)|\vee|g(t,x)|\vee|Lg(t,x)|\leq M(1+|x|^{\alpha+1}),$

(2)

where $M$ depends on $C$ and $\sup_{0\leq t\leq T}\left(|f(t,0)|+|g(t,0)|+|Lg(t,0)|\right)$.

Similar to the relation between Assumption 1 and (2), Assumption 3 can be derived from Assumption 2 but with complicated relations between $p$ and $q$ as well as $K$ and $K_{1}$. So we present Assumption 3 as a new assumption.

Now we turn to the requirement on the temporal variables in the coefficients.

Now we introduce the procedure of constructing the Milstein-type method discussed in this paper.

Step 1. Based on the formates of the coefficients, we choose a strictly increasing continuous function $\mu:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that $\mu(u)\rightarrow\infty$ as $u\rightarrow\infty$ and for any $l=1,2,...,d$.

$\displaystyle\sup_{0\leq t\leq T}\sup_{|x|\leq u}(|f(t,x)|\vee|g(t,x)|\vee|G^{l}(t,x)|)\leq\mu(u),\quad u\geq 1.$

Step 2. We choose a constant $\hat{\kappa}\geq 1\wedge\mu(1)$ and a strictly decreasing function $\kappa:(0,1]\rightarrow[\mu(1),\infty)$ such that

$\displaystyle h^{1/4}\kappa(h)\leq\hat{\kappa}\quad\text{for any}~{}h\in(0,1]\quad\text{and}\quad\lim_{h\rightarrow 0}\kappa(h)=\infty.$

(3)

Step 3. Since the inverse function of $\mu$, denoted by $\mu^{-1}$, is a strictly increasing continuous function from $[\mu(0),\infty)$ to $\mathbb{R}_{+}$, for a given step size $h\in(0,1]$ we define the truncated mapping by

$\displaystyle\pi_{h}(x)=\left(|x|\wedge\mu^{-1}(\kappa(h))\right)\frac{x}{|x|},$

where $x/|x|$ is set to be 0 if $x=0$. Then we define the truncated functions by

$\displaystyle f_{h}(t,x)=f(t,\pi_{h}(x)),\quad g_{h}(t,x)=g(t,\pi_{h}(x)),\quad G^{l}_{h}(t,x)=G^{l}(t,\pi_{h}(x)).$

for any $x\in\mathbb{R}^{d}$ and $l=1,2,...,d$. It is not hard to see that for any $t\in[0,T]$ and any $x\in\mathbb{R}^{d}$,

$\displaystyle|f_{h}(t,x)|\vee|g_{h}(t,x)|\vee|G^{l}_{h}(t,x)|\leq\mu(\mu^{-1}(\kappa(h)))=\kappa(h).$

(4)

we can also obtain the fact that there exists a positive constant $\hat{M}$ such that

$\displaystyle|\frac{\partial f_{h}(t,x)}{\partial x}|\vee|\frac{\partial^{2}f_{h}(t,x)}{\partial x^{2}}|\vee|\frac{\partial g_{h}(t,x)}{\partial x}|\vee|\frac{\partial^{2}g_{h}(t,x)}{\partial x^{2}}|\leq\hat{M},$

for any $t\in[0,T]$ and all $x\in\mathbb{R}^{d}$.

Step 4. Now we turn to discretise the process $E(t)$ in a finite time interval $[0,T]$ for any given $T>0$. For the given step size h, set $t_{i}=ih$ and let $\Delta_{i}$ be independently identically sequence satisfying $\Delta_{i}=D(h)$ in distribution for $i=0,1,2,...$. By the iteration, $D_{h}(t_{i})=D_{h}(t_{i-1})+\Delta_{i}$ with $D_{h}(0)=0$, the sample path of $D(t)$ can be simulated. And we stop the iteration for some positive integer $N$ when

$\displaystyle T\in[D_{h}(t_{N}),D_{h}(t_{N+1}))$

holds.

Step 5. The discretised $E(t)$, denoted by $E_{h}(t)$, can be found by

$\displaystyle E_{h}(t)=\big{(}\min\{n;D_{h}(t_{n})>t\}-1\big{)}h,$

(5)

for $t\in[0,T]$. It is not hard to see $E_{h}(t)=ih$ for $t\in\left[D_{h}(t_{i}),D_{h}(t_{i+1})\right)$.

For $i=0,1,2,...,N$, denote $\tau_{i}=D_{h}(t_{i})$. Then it can be observed that

$\displaystyle E_{h}(\tau_{i})=E_{h}(D_{h}(t_{i}))=ih.$

(6)

Step 6. Finally by setting $X_{0}=Y(0)$, the discrete version of the Milstein method is defined as

	$\displaystyle X_{\tau_{n+1}}=$	$\displaystyle X_{\tau_{n}}+f_{h}(\tau_{n},X_{\tau_{n}})\bigg{(}E_{h}(\tau_{n+1})-E_{h}(\tau_{n})\bigg{)}$
		$\displaystyle+g_{h}(\tau_{n},X_{\tau_{n}})\bigg{(}W(E_{h}(\tau_{n+1}))-W(E_{h}(\tau_{n}))\bigg{)}$
		$\displaystyle+\dfrac{1}{2}\sum_{l=1}^{d}g^{l}_{h}(\tau_{n},X_{\tau_{n}})G^{l}_{h}(\tau_{n},X_{\tau_{n}})\bigg{(}\Delta W^{2}(E_{h}(\tau_{n}))-\Delta(E_{h}(\tau_{n}))\bigg{)}.$		(7)

It should be noted that $\{\tau_{n}\}_{n=1,2,...,N}$ is a random sequence but independent from the Wiener process. In addition, it is not hard to see from (6) that

$\displaystyle E_{h}(\tau_{n+1})-E_{h}(\tau_{n})=h~{}~{}\text{and}~{}~{}W(E_{h}(\tau_{n+1}))-W(E_{h}(\tau_{n}))=W((n+1)h)-W(nh).$

Now we present the continuous version of (2), as it is more convenient to use it in our proofs.

For any $t\in[0,T]$ and any $x\in\mathbb{R}^{d}$, set

$\displaystyle Lg_{h}(t,x):=\sum_{l=1}^{d}g_{h}^{l}(t,x)G_{h}^{l}(t,x),$

For any $t\in[0,T]$. the continuous version of our Milstein method is

	$\displaystyle X(t)=$	$\displaystyle X(0)+\int_{0}^{t}f_{h}{(\bar{\tau}(s),\bar{X}(s))dE(s)}+\int_{0}^{t}g_{h}{(\bar{\tau}(s),\bar{X}(s))dW(E(s))}$
		$\displaystyle+\int_{0}^{t}Lg_{h}{(\bar{\tau}(s),\bar{X}(s))\Delta W(E_{h}(s))dW(E(s))},$		(8)

where $\bar{\tau}(s)=\tau_{n}\mathbf{1}_{[\tau_{n},\tau_{n+1})}(s)$, $\bar{X}(t)=\sum_{n=0}^{N}X_{\tau_{n}}\mathbf{1}_{[\tau_{n},\tau_{n+1})}(t)$ and

$\displaystyle\Delta W(E_{h}(s))=\sum_{i=1}^{N}\mathbf{1}_{\left\{\tau_{i}\leqslant s<\tau_{i+1}\right\}}\big{(}W(E_{h}(s))-W(E_{h}(\tau_{i}))\big{)}.$

The following version of the Taylor expansion is essential for proofs in our paper.

Given $\psi:\mathbb{R}^{d+1}\to\mathbb{R}^{d}$ is a third-order continuously differentiable function, for $z,z^{*}\in\mathbb{R}^{d+1}$ we have

$\displaystyle\begin{split}\psi(z)-\psi(z^{*})=\psi^{{}^{\prime}}(z)|_{z=z^{*}}(z-z^{*})+R_{\psi}(z,z^{*}),\end{split}$

where

$\displaystyle\begin{split}R_{\psi}(z,z^{*})=\int_{0}^{1}(1-\theta)\psi^{{}^{\prime\prime}}(z)|_{z=z^{*}+\theta(z-z^{*})}(z-z^{*},z-z^{*})d\theta.\end{split}$

Here, $\psi^{{}^{\prime}}$ and $\psi^{{}^{\prime\prime}}$are defined in the following way, for any $z,\hat{z},\tilde{z}\in\mathbb{R}^{d+1}$.

$\displaystyle\psi^{{}^{\prime}}(z)(\hat{z})=\sum_{i=1}^{d+1}\dfrac{\partial\psi}{\partial z^{i}}\hat{z}_{i},\quad\psi^{{}^{\prime\prime}}(z)(\hat{z},h)=\sum_{i,k=1}^{d+1}\dfrac{\partial^{2}\psi}{\partial z^{i}\partial z^{k}}\hat{z}_{i}\tilde{z}_{k},$

where $\psi=(\psi_{1},\psi_{2},...,\psi_{d})^{T}$, $\psi_{j}:\mathbb{R}^{d+1}\to\mathbb{R}$ for $j=1,2,...d$, and $\dfrac{\partial\psi}{\partial z^{i}}=(\dfrac{\partial\psi_{1}}{\partial z^{i}},\dfrac{\partial\psi_{2}}{\partial z^{i}},...,\dfrac{\partial\psi_{d}}{\partial z^{i}})^{T}$ for $i=1,2,...,d+1$.

In the paper, we employ the Taylor expansion above by using one dimension for the time variable and $d$ dimensions for the state variable, to be more precise, we set $z=(\eta,\bar{x})$ and $z^{*}=(\eta,x^{*})$ for $\eta\in\mathbb{R}_{+}$ and $\bar{x},x^{*}\in\mathbb{R}^{d}$. It is clear that $z-z^{*}=(0,\bar{x}-x^{*})$, in the case. Therefore, for $\psi:\mathbb{R}_{+}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ we have that

$\displaystyle\begin{split}\psi(\eta,\bar{x})-\psi(\eta,x^{*})=\psi^{{}^{\prime}}(\eta,x)\big{|}_{x=x^{*}}(\bar{x}-x^{*})+R_{\psi}(\eta,\bar{x},x^{*}),\end{split}$

where

$\displaystyle\begin{split}R_{\psi}(\eta,\bar{x},x^{*})=\int_{0}^{1}(1-\theta)\psi^{{}^{\prime\prime}}(\eta,x)|_{x=x^{*}+\theta(\bar{x}-x^{*})}(\bar{x}-x^{*},\bar{x}-x^{*})d\theta,\end{split}$

for any $\eta\in\mathbb{R}_{+}$ and $\bar{x},x^{*}\in\mathbb{R}^{d}$. In this case, for any $x,\bar{j},\bar{h}\in\mathbb{R}^{d},\psi^{{}^{\prime}}$ and $\psi^{{}^{\prime\prime}}$ are defined by

$\displaystyle\psi^{{}^{\prime}}(\eta,x)(\bar{j})=\sum_{i=1}^{d}\dfrac{\partial\psi}{\partial x^{i}}\bar{j}_{i},\quad\psi^{{}^{\prime\prime}}(\eta,x)(\bar{j},\bar{h})=\sum_{i,k=1}^{d}\dfrac{\partial^{2}\psi}{\partial x^{i}\partial x^{k}}\bar{j}_{i}\bar{h}_{k}.$

respecttively. Here, $\psi=(\psi_{1},\psi_{2},...,\psi_{d})^{T}$,$\dfrac{\partial\psi}{\partial x^{i}}=(\dfrac{\partial\psi_{1}}{\partial x^{i}},\dfrac{\partial\psi_{2}}{\partial x^{i}},...,\dfrac{\partial\psi_{d}}{\partial x^{i}})^{T}$, $\bar{j}=(\bar{j}_{1},\bar{j}_{2},...,\bar{j}_{d})^{T}$ and $\bar{h}=(\bar{h}_{1},\bar{h}_{2},...,\bar{h}_{d})^{T}$.

Setting $\eta=\bar{\tau}(t),\bar{x}=X(t)$ and $x^{*}=\bar{X}(t)$, we derive from above, that for any fixed $t\in[0,T]$,

	$\displaystyle\psi(\bar{\tau}(t),X(t))-\psi(\bar{\tau}(t),\bar{X}(t))=$	$\displaystyle\psi^{{}^{\prime}}(\bar{\tau}(t),x)\big{|}_{x=\bar{X}(t)}\int_{0}^{t}g_{h}(\bar{\tau}(s),\bar{X}(s))dW(E(s))$
		$\displaystyle+\tilde{R}_{\psi}(t,X(t),\bar{X}(t)),$		(9)

Here

	$\displaystyle\tilde{R}_{\psi}(t,X(t),\bar{X}(t))=$	$\displaystyle\psi^{{}^{\prime}}(\bar{\tau}(t),x)\big{|}_{x=\bar{X}(t)}\bigg{(}\int_{0}^{t}f_{h}(\bar{\tau}(s),\bar{X}(s))dE(s)$
		$\displaystyle+\int_{0}^{t}Lg_{h}(\bar{\tau}(s),\bar{X}(s))\Delta W(E_{h}(s))dW(E(s))\bigg{)}$
		$\displaystyle+R_{\psi}(\bar{\tau}(t),X(t),\bar{X}(t)).$		(10)

Thus, resplacing $\psi$ by $g_{h}$, we obtain

		$\displaystyle\tilde{R}_{g_{h}}(t,X(t),\bar{X}(t))$
	$\displaystyle=$	$\displaystyle g_{h}(\bar{\tau}(t),X(t))-g_{h}(\bar{\tau}(t),\bar{X}(t))-Lg_{h}(\bar{\tau}(t),\bar{X}(t))\Delta W(E_{h}(t)).$		(11)

At the end of this section, we mention some known results. For the proofs of Lemmas 1 and 2, we refer the readers to [10]. The proof of Lemma 3 can be found in [15]. Lemma 4 is borrowed from [19].

Briefly speaking, Lemmas 1 and 2 indicate that, to some extended, the truncated functions $f_{h}$ and $g_{h}$ inherit Assumptions 1 and 3. Lemma 3 is useful for the analysis of the convergence order of $E_{h}(t)$. Lemma 4 states the moment boundedness of the underlying solution.

Lemmas that will be used in the proofs of main results in Section 4 are presented and proved in this section.

Proof. Fix any $h\in(0,1]$, $\hat{p}>2$ and $t\geq 0$. There is a unique integer $n\geq 0$ such that $\tau_{n}\leq t<\tau_{n+1}$. By properties of the basic inequality, we then derive from (2) that

		$\displaystyle|X(t)-\bar{X}(t)|^{\hat{p}}$
	$\displaystyle=$	$\displaystyle|X(t)-X(\tau_{n})|^{\hat{p}}$
	$\displaystyle=$	$\displaystyle\bigg{|}\int_{\tau_{n}}^{t}{f_{h}(\bar{\tau}(s),\bar{X}(s))dE(s)}+\int_{\tau_{n}}^{t}{g_{h}(\bar{\tau}(s),\bar{X}(s))dW(E(s))}$
		$\displaystyle+\int_{\tau_{n}}^{t}{Lg_{h}(\bar{\tau}(s),\bar{X}(s))\Delta W(E_{h}(s))dW(E(s))}\bigg{|}^{\hat{p}}$
	$\displaystyle\leq$	$\displaystyle 3^{\hat{p}-1}\bigg{(}\left|\int_{\tau_{n}}^{t}{f_{h}(\bar{\tau}(s),\bar{X}(s))dE(s)}\right|^{\hat{p}}+\left|\int_{\tau_{n}}^{t}{g_{h}(\bar{\tau}(s),\bar{X}(s))dW(E(s))}\right|^{\hat{p}}$
		$\displaystyle+\left|\int_{\tau_{n}}^{t}{Lg_{h}(\bar{\tau}(s),\bar{X}(s))\Delta W(E_{h}(s))dW(E(s))}\right|^{\hat{p}}\bigg{)}.$		(14)

Now, we estimate those three terms inside the bracket on the right hand side of the last inequality of (3) By the Hölder inequality, the first term can be estimated by

$\displaystyle\mathbb{E}_{W}\left|\int_{\tau_{n}}^{t}{f_{h}(\bar{\tau}(s),\bar{X}(s))dE(s)}\right|^{\hat{p}}\leq h^{\hat{p}-1}\mathbb{E}_{W}\int_{\tau_{n}}^{t}{\left|f_{h}(\bar{\tau}(s),\bar{X}(s))\right|^{\hat{p}}dE(s)}.$

(15)

The second item Let $x(t)=\int_{\tau_{n}}^{t}{g_{h}(\bar{\tau}(s),\bar{X}(s))dW(E(s))}$, so we have

		$\displaystyle\mathbb{E}_{W}\left|x(t)\right|^{\hat{p}}$
	$\displaystyle=$	$\displaystyle\dfrac{\hat{p}}{2}\mathbb{E}_{W}\int_{\tau_{n}}^{t}\left(\left|x(s)\right|^{\hat{p}-2}\big{|}{g_{h}(\bar{\tau}(s),\bar{X}(s))\big{|}^{2}+({\hat{p}-2})\left|x(s)\right|^{\hat{p}-4}}\big{|}x^{T}(s)g(s)\big{|}^{2}\right)dE(s)$
	$\displaystyle\leq$	$\displaystyle\frac{\hat{p}(\hat{p}-1)}{2}\mathbb{E}_{W}\int_{\tau_{n}}^{t}\left|x(s)\right|^{\hat{p}-2}\left|g_{h}(\bar{\tau}(s),\bar{X}(s))\right|^{2}dE(s)$
	$\displaystyle\leq$	$\displaystyle\frac{\hat{p}(\hat{p}-1)}{2}\left(\mathbb{E}_{W}\int_{\tau_{n}}^{t}\left|x(s)\right|^{\hat{p}}dE(s)\right)^{\frac{\hat{p}-2}{\hat{p}}}\left(\mathbb{E}_{W}\int_{\tau_{n}}^{t}\left|g_{h}(\bar{\rho}(s),\bar{X}(s))\right|^{\hat{p}}dE(s)\right)^{\frac{2}{\hat{p}}}$
	$\displaystyle=$	$\displaystyle\frac{\hat{p}(\hat{p}-1)}{2}\left(\int_{\tau_{n}}^{t}\mathbb{E}_{W}\left|x(s)\right|^{\hat{p}}dE(s)\right)^{\frac{\hat{p}-2}{\hat{p}}}\left(\mathbb{E}_{W}\int_{\tau_{n}}^{t}\left|g_{h}(\bar{\rho}(s),\bar{X}(s))\right|^{\hat{p}}dE(s)\right)^{\frac{2}{\hat{p}}}.$