As for the investigation of stochastic differential equations(SDEs) on solutions and system stabilities, the comparison theorems have been evolving not only as very powerful tools but also nice train of thought. Abundant contributions on comparison theorems have been already concerned with different random models or different stochastic resources, for more detail we refer to [1-6] and the reference therein.

In 1970s, Doss [7], Sussmann [8], Krasnoselski and Pokrovski [9] gave a new method that convert SDE to ODE with parameter. Huang [10] gave the definition of generalized sample solution of more general SDE. Jiang [11] also obtained the generalized sample solution of Volterra integral equations. Applying directly the theory of ODE, this method was very valid whether in the theory of SDE or in the numerical calculus of SDE. Inspired by these, we engage to study the comparison theorem of generalized sample solution for stochastic differential equation (SDE) with different drift and diffusion coefficients which are driven by fractional Brownian motions (fBm). That is

$\begin{equation}\label{main} X_t=x_0+\int_{t_0}^{t}f(s,B_s^H,X_s)ds+\int_{t_0}^{t}\sigma(s,B_s^H,X_s)dB_s^H, \ \ t\geq t_0, \end{equation}$

(1.1)

where $B_t^H$ is one dimensional fBm on a suitable complete probability space $(\Omega, \mathcal {F},\mathcal {F}_t,P)$, $H(\geq\frac{1}{2})$ is the Hurst parameter, and the stochastic integral throughout the article is taken as pathwise integral in the sense of [12].

In terms of fractional Brownian motion $B^{H}_{t}$ with Hurst parameter $H\in(0,1)$, it is a continuous-time Gaussian process with mean zero, starting at zero and having the following correlation function

$ E[B^{H}_{s}B^{H}_{t}]=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}). $

Particularly, it is not semimartingale nor Markovian process, but when $H=\frac{1}{2}$ it reduces to the standard Brownian motion. Since its' self-similarity and long-range dependence, fBm has been a driving noise in models arising in physics, telecommunication network, finance and other fields and give rise to much more research interest recently. Numerous results on fBm have appeared, such as [13, 14] and so on.

The paper is organized as follows: in Section 2, we recall some basic results on the SDE. In Section 3, we characterize the generalized sample solution and show the comparison theorem for the solution of SDE driven by fractional Brownian motions. In Section 4, we give an examples of the comparison theorem and propose an asymptotic optimal strategy for the solutions of SDEs.

In this section, we recall some important results of SDE which are used in the following contents. (For the detail, we refer to [14])

Lemma 2.1 { Suppose the following SDE with one dimensional fBm $B_t^H$ ($H\in(\frac{1}{2},1)$) defined on a complete probability space $(\Omega, \mathcal {F}, \mathcal {F}_t, P)$,

$\begin{equation}\label{sdeexist} X_t=X_0+\int_0^{t}f(s,X_s)\textrm{d}s+\int_0^{t}\sigma(s,X_s)\textrm{d}B_s^H, t\in[0,T], \end{equation}$

(2.1)

where the function $\sigma=\sigma(t,x):[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ satisfies the assumptions: $\sigma$ is differentiable in $x$, there exist $M>0, 0<\gamma, \kappa\leq1$ and for any $N>0$ there exists $M_N$ such that

(ⅰ) $\sigma$ is Lipschitz continuous in $x$:

$ |\sigma(t,x)-\sigma(t,y)|\leq{M|x-y|}, \forall {t\in[0,T]} x,y\in\mathbb{R}; $

(ⅱ) $x$-derivative of $\sigma$ is local Hölder continuous in $x$:

$ |\sigma_x(t,x)-\sigma_x(t,y)|\leq{M_N|x-y|^\kappa}, \forall |x|,|y|\leq{N} {t\in[0,T]}; $

(ⅲ) $\sigma$ is Hölder continuous in time $t$:

$ |\sigma(t,x)-\sigma(s,x)|+|\sigma_x(t,x)-\sigma_x(s,x)|\leq{M|t-s|^\gamma} \forall x\in\mathbb{R}, {t,s\in[0,T]}. $

The function $f=f(t,x):[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following assumptions:

(ⅳ) for any ${N}\geq0$, there exists $C_N$ such that

$ |f(t,x)-f(t,y)|\leq{C_N|x-y|}, \forall |x|,|y|\leq{N} {t,s\in[0,T]}; $

(Ⅴ) there exists a function $f_0\in{L_p[0,T]}$ and $C>0$ such that

$ |f(t,x)|\leq{C|x|}+f_0(t) , \forall (t,x)\in[0,T]\times\mathbb{R}. $

And $p=(1-H+\epsilon)^{-1}$ with some $0<\epsilon<H-\frac{1}{2}, \gamma>{1-H}, \kappa>H^{-1}-1$ (the constants $M$, $M_N$, $N$, $C_N$ and the function $f_0$ can depend on $\omega$). Then there exists an unique solution $\{X_t ; t\in[0,T]\} $ of equation (2.1).}

Lemma 2.2 { For $\alpha>1-H$, let $a(s)\in{C^\alpha[0,t]}$ and $v(s)\in{L^1([0,T])}$. Define

$ Y_t=Y_0+\int_0^{t}v(s)ds+\int_0^{t}a(s)dB_s^{H},\ \ H\in(\frac{1}{2},1), $

moreover, assume $F=F(t,x):\mathbb{R}_+\times\mathbb{R}\rightarrow\mathbb{R}$, $F\in{C^1(\mathbb{R}_+)}\times{C^2(\mathbb{R})}$ such that $F_x^\prime(s,Y_s)a(s)\in{C^\beta[0,t]}$ for $\beta+H>1$ a.s.. Then

$\begin{align} F(t,Y_t)=&F(0,Y_0)+\int_0^{t}F_t^\prime(s,Y_s)\textrm{d}s+\int_0^{t}F_x^\prime(s,Y_s)v(s)\textrm{d}s \notag \\ &+\int_0^{t}F_x^\prime(s,Y_s)a(s)\textrm{d}B_s^{H}. \notag \ \ \ \end{align}$

In particular, for $F\in C^2(\mathbb{R})$, we have

$ F(B_t^H)=F(0)+\int_0^{t}F^\prime(B_s^H)\textrm{d}B_s^{H} . $

Lemma 2.3  Assume $f(t,x)$, $\tilde{f}(t,x)$ satisfy Carathéodory conditions on $G^\star\subset\mathbb{R}^2$(i.e. they are measurable in $t$, continuous in $x$ and dominated by a locally integrable function $m(t)$ in the domain $G^\star$). Let $(t_0, x_0)\in G^\star$, $(t_0, \tilde{x}_0)\in G^\star$ be two points such that $x_0\leq\tilde{x}_0$, and $x(t)$ be any solution to the initial value problem $ \dot{x}_t=f(t,x_t), x(t_0)=x_0; $ $\tilde{x}(t)$ be the maximal solution [15] to the problem $ \dot{x}_t=\tilde{f}(t,x_t), x(t_0)=\tilde{x}_0. $ Moreover,

$ (t-t_0)f(t,x)\leq(t-t_0)\tilde{f}(t,x) $

holds in $G^\star$. Then $x(t)\leq\tilde{x}(t)$ in the interval of both them exist.

In what follows, we consider the SDE (1.1) driven by one dimensional fBm $B_t^H$ with $H\in(\frac{1}{2},1)$ on a suitable complete probability space $(\Omega,\mathcal{F},\mathcal{F}_t,P)$ and $\{\mathcal {F}_t, t\geq t_0 \}$ an increasing family of subalgebras of $\mathcal {F}$ satisfying the usual conditions. The coefficients satisfy the following conditions $(\mathbb{H}_1)$ and $(\mathbb{H}_2)$:

$(\mathbb{H}_1)$ $f=f(t,b,x)$, $\sigma=\sigma(t,b,x)$ are two real valued functions defined on $G\subset[t_0, \infty)\times\mathbb{R}^{2}$;

$(\mathbb{H}_2)$ $\sigma\in C^2(G)$ and $f\in C^0(G)$ and locally Lipschitz continuous with respect to $x$ in $G$.

For detailed study on the well-defined SDE (1.1), we refer to [14, 16].

Now, we adopt three steps to characterize the generalized sample solution of SDE (1.1). First, we consider the following deterministic initial value problem:

$\begin{equation} \label{ode1} \frac{\textrm{d}y}{\textrm{d}b}=\sigma(t,b,y) , \ \ y(b_0)=\xi, \end{equation}$

(3.1)

where $t$ is a parameter. Since $\sigma\in C^2(G)$, by common theories of ordinary differential equation, we imply that equation (3.1) have an unique $C^2$-solution $y=\Phi(t,b,\xi,b_0)$ in some domain $\tilde{G}\subset[t_0, \infty)\times\mathbb{R}^2$. Denote

$\begin{equation*} h(t, \xi, \omega)= \frac{f\left(t, B_t^H(\omega), \Phi\left(t, B_t^H(\omega), \xi, b_0\right)\right) -\Phi_t\left(t, B_t^H(\omega), \xi, b_0\right)}{\Phi_{\xi}\left(t, B_t^H(\omega), \xi, b_0)\right)}, \end{equation*}$

where $\Phi_t$, $\Phi_{\xi}$ indicate partial derivatives with the variables $t$ and $\xi$, respectively.

Second, we consider the following initial value problem

$\begin{equation}\label{ode2} \frac{\textrm{d}\xi}{\textrm{d}t}=h(t,\xi,\omega) , \ \ \xi(t_0)=x_0(\omega). \end{equation}$

(3.2)

Due to the function $f(t,b,x)$ is locally Lipschitz continuous with respect to $x$ in $G$, the equation (3.2) exists an unique solution $\xi(t,\omega),\ \ t_0\leq t <\zeta(\omega),$ where $\zeta$ is the ``explosion time".

Finally, we intend to show that $X(t,\omega)=\Phi(t,B_t^H(\omega),\xi(t,\omega),b_0)$ is just the unique solution of equation (1.1).

Theorem 3.1 { Assuming that the drift and diffusion coefficient $f$, $\sigma$ satisfy the foregoing conditions $(\mathbb{H}_1)$ and $(\mathbb{H}_2)$, there exists an unique strong solution for SDE (1.1),

$\begin{equation}\label{main2} X(t,\omega)=\Phi(t,B_t^H(\omega),\xi(t,\omega),b_0) , \ \ t_0\leq t <\zeta(\omega) , \end{equation}$

(3.3)

where $\Phi$ and $\xi$ are the solutions to the problems (3.1) and (3.2), respectively, and $\zeta$ is the "explosion time" for $X_t$}.

Proof Since $\xi(t,\omega)$ is an adapted continuous process, we could find a sequence of compact sets $ K_n $ in $G$ such that $K_n\uparrow G$ and the well-defined stopping time

$ \zeta_n(\omega)=\inf\left\{{t>0 : \left(t,B_t^H(\omega),\Phi\left(t,B_t^H(\omega),\xi(t,\omega),b_0\right)\right)\notin K_n}\right\}, $

which converges to $\zeta$ almost surely. Therefore, by applying Lemma 2.2 to the process

$ X(t\wedge\zeta)=\Phi(t\wedge\zeta,B_t^H(\omega),\xi(t\wedge\zeta),b_0), $

we have

$\begin{split} X(t\wedge\zeta)- x_0&= \int_{t_0}^{t\wedge\zeta}\Phi_b(s,B_s^H,\xi_s,b_0)\textrm{d}B_s^H +\int_{t_0}^{t\wedge\zeta}\Phi_t\textrm{d}s+\int_{t_0}^{t\wedge\zeta}\Phi_\xi\textrm{d}\xi_s \\ &=\int_{t_0}^{t\wedge\zeta}\sigma(s,B_s^H,X_s)\textrm{d}B_s^H+\int_{t_0}^{t\wedge\zeta}\Phi_t\textrm{d}s +\int_{t_0}^{t\wedge\zeta}\Phi_\xi{h(s,\xi_s,\omega)}\textrm{d}s \\ &=\int_{t_0}^{t\wedge\zeta}\sigma(s,B_s^H,X_s)\textrm{d}B_s^H+\int_{t_0}^{t\wedge\zeta}f(s,B_s^H,X_s)\textrm{d}s, \ \ \ \ \ \ {\rm a.s.} , \end{split}$

which shows that $ \{ X_t , 0\leq t < \zeta \} $ is a solution to SDE (1.1).

Conversely, if $ \{\tilde{X}_t , 0\leq t < \tilde{\zeta} \} $ is an another solution to (1.1), we can also use Lemma 2.2 to the process

$ \tilde{\xi}(t\wedge\tilde{\zeta})\equiv\Phi(t\wedge\tilde{\zeta},\tilde{X}(t\wedge\tilde{\zeta}),B_{t\wedge\tilde{\zeta}}^H,b_0) $

to verify similarly that $ \{ \tilde{\xi}(t) , 0\leq t < \tilde{\zeta} \} $ is a solution to problem (3.2)(cf [2]). It follows from the uniqueness of the solution to problem (3.2) that $\tilde{\zeta}\leq\zeta$ and $\tilde{\xi}(t)$ coincides with $\xi(t)$ in the interval $[0 ,\tilde{\zeta}) $. Therefore,

$\begin{split} \tilde{X}(t\wedge\tilde{\zeta}) &=\Phi(t\wedge\tilde{\zeta},B_{t\wedge\tilde{\zeta}}^H,\tilde{\xi}(t\wedge\tilde{\zeta}),b_0)\\ &=\Phi(t\wedge\tilde{\zeta},B_{t\wedge\tilde{\zeta}}^H,\xi(t\wedge\tilde{\zeta}),b_0) =X(t\wedge\tilde{\zeta}), \ \ \ \ {\rm a.s.}, \end{split}$

which completes the proof.

Remark 3.1 When the solutions of problems (3.1) and (3.2) exist, we call (3.3) is the generalized sample solution of SDE (1.1). For more detail on generalized sample solution, see [10, 11].

With the definition of generalized sample solution and the above lemmas, we now illustrate the main results on comparison theorems.

Theorem 3.2  Assume that the drift and diffusion coefficients $f$ and $\sigma$ of SDE (1.1) satisfy the conditions of Theorem 3.1. Moreover,

(ⅰ) there exists a function $\tilde{\sigma}=\tilde{\sigma}(t,b,x)$ defined on $G$ satisfying the Carathéodory conditions in $(b,x)$ (i.e., they are measurable in $t$, continuous in $(b,x)$ and dominated by a locally integrable function $\tilde{m}(t)$ in the domain $G$) and the inequality

$\begin{equation}\label{inequa1} (b-b_0)\sigma(t,b,x)\leq (b-b_0)\tilde{\sigma}(t,b,x) {\rm in} G; \end{equation}$

(3.4)

(ⅱ) there exists a function $\tilde{h}$ defined on $D$ which is the domain of $h$, satisfying the Carathéorory conditions in $(t,\xi)$ and the inequality

$\begin{equation}\label{inequa2} h(t,\xi,\omega)\leq\tilde{h}(t,\xi,\omega) {\rm a.e.} \omega {\rm and} (t,\xi)\in D; \end{equation}$

(3.5)

(ⅲ) denoted by $\tilde{\Phi}$ and $\tilde{\xi}$ are the maximal solutions to the following two problems

$\begin{equation}\label{problem1} \frac{\textrm{d}y}{\textrm{d}b}=\tilde{\sigma}(t,b,y) , y(t_0)=\xi^\star; \end{equation}$

(3.6)

$\begin{equation}\label{problem2} \frac{\textrm{d}\xi}{\textrm{d}t}=\tilde{h}(t,\xi,\omega) , \xi(t_0)=\tilde{x}_0(\omega), \end{equation}$

(3.7)

respectively, and $x_0\leq\tilde{x}_0$. Then for the unique solution $X_t$ of SDE (1.1)

$ X(t)\leq\tilde{\Phi}(t,B_t^H(\omega),\tilde{\xi}(t,\omega),b_0) $

holds for a.e. $\omega$ and all $t$ on the common interval where both sides are well-defined. }

Proof  Using Lemma 2.3, we have

$ \Phi(t,b,\xi,b_0)\leq \tilde{\Phi}(t,b,\xi^\star,b_0), $

provided $\xi\leq\xi^\star$. By the same reason and $x_0\leq\tilde{x}_0$, we could also show that $\xi(t,\omega)\leq\tilde{\xi}(t,\omega)$ on the common interval where both sides are well-defined.

According to these two inequalities and Theorem 3.1, we obtain

$ X(t,\omega)=\Phi(t, B_t^H(\omega), \xi(t,\omega),b_0)\leq\tilde{\Phi}(t, B_t^H(\omega), \tilde{\xi}(t,\omega),b_0). $

Remark 3.2 The conditions imposed on $\sigma$ in Theorem 3.2 would be too restrictive, we can weaken them by a method of smooth approximation and obtain a similar result.

For any $\epsilon>0$, set $\varphi_\epsilon$ be a modified on $\mathbb{R}^3$, that is, $\varphi_\epsilon\in{C_0^\infty(\mathbb{R}^3)}$, $\varphi_\epsilon\geq0$, $\int\varphi_\epsilon=1$ and the support of $\varphi_\epsilon$ is contained in the ball $U(0,\epsilon)$. If necessary, we can extend its domain and assume that $\sigma$ is well-defined and continuous in $\mathbb{R}^3$. Let $\sigma^{(\epsilon)}= \sigma\ast\varphi_\epsilon$, where $\ast$ denotes the convolution. Hence, $\sigma^{(\epsilon)}\in{C^\infty(\mathbb{R}^3)}$ and $\sigma^{(\epsilon)}$ converges to $\sigma$ uniformly on every compact subset in $G$ as $\epsilon\rightarrow 0$.

Let $\Phi^{(\epsilon)}$ be the unique solution to the problem

$ \frac{\textrm{d}\Phi}{\textrm{d}b}=\sigma^{(\epsilon)}(t,b,\Phi) , \Phi(b_0)=\xi; $

and \begin{equation*} h^{(\epsilon)}(t, \xi, \omega)\equiv\frac{f\left(t, B_t^H(\omega), \Phi^{(\epsilon)}(t, B_t^H(\omega), \xi, b_0)\right)-\Phi^{(\epsilon)}_t\left(t, B_t^H(\omega), \xi, b_0\right)} {\Phi^{(\epsilon)}_{\xi}\left(t, B_t^H(\omega), \xi, b_0\right)}. \end{equation*}

Hence, we can obtain the following result.

Theorem 3.3 { Assume that the drift and diffusion coefficients $f(t,b,x)$ and $\sigma(t,b,x)$ of SDE (1.1) are continuous in $G$ and locally Lipschitz continuous w.r.t $x$. Let $X(t)$ and $\Phi(t,b,\xi,b_0)$ be the unique solutions to SDE (1.1) and problem (3.1) respectively. Moreover, there exist two functions $\tilde{\sigma}$ satisfying inequality (3.4) and $\tilde{h}$ satisfying

$ h^{(\epsilon)}(t,\xi,\omega)\leq\tilde{h}(t,\xi,\omega) (t,\xi,\omega)\in D $

for any sufficiently small positive number $\epsilon$. Then

$ X(t)\leq\tilde{\Phi}(t,B_t^H(\omega),\tilde{\xi}(t,\omega),b_0) $

holds for a.e. $\omega$ and all $t$ on the common interval where both sides are well-defined, where $X_t$ is the unique solution of SDE (1.1) and $\tilde{\Phi}$, $\tilde{\xi}$ are the maximal solutions to (3.6), (3.7).

Proof  The existence and uniqueness of solution $X(t)$ to SDE (1.1) are easily obtained. Denoted by $\varsigma$ its ``explosion time" and choose a sequence of compact sets $\{K_n\}$ such that $K_n\uparrow{G}$. Define

$ \varsigma_n(\omega)=\inf\left\{{t>t_0 : \left(t,B_t^H(\omega),X(t,\omega)\right)\notin K_n}\right\}, $

then each $\varsigma_n$ is a stopping time and $\varsigma_n\uparrow\varsigma$. Since $f$ and $\sigma^{(\epsilon)}$ satisfy the conditions stated in Theorem 3.2, and for $\epsilon>0$ sufficiently small, the solution $X^{(\epsilon)}(t)$ to the SDE

$ X_t=x_0+\int_{t_0}^{t}f(s,B_s^H,X_s)\textrm{d}s+\int_{t_0}^{t}\sigma^{(\epsilon)}(s,B_s^H,X_s)\textrm{d}B_s^H, \ \ \ \ \ \ \ \ \ t\geq{t_0}, $

satisfies the inequality

$\begin{equation}\label{apposolution} X^{(\epsilon)}(t)\leq\tilde{\Phi}(t,B_t^H(\omega),\tilde{\xi}(t,\omega),b_0). \end{equation}$

(3.8)

For every sequence $\{\epsilon_m\}$ such that $\epsilon_m\downarrow0$ and every $n$, we have

$ \lim\limits_{m\to\infty}E\left\{\sup\limits_{t\leq\varsigma_n}\mid{X^{(\epsilon_m)}(t)-X(t)}\mid^2\right\}=0. $

Hence, there exists a subsequence, again denoted by $\{\epsilon_m\}$, such that

$ P\left\{{\lim\limits_{m\to\infty}{\sup\limits_{t\leq\varsigma_n}\mid{X^{(\epsilon_m)}(t)-X(t)}\mid}=0}\right\}=1. $

Upon passage to limit in (3.8), the proof is complete.

In the section, we present an examples of the comparison theorem and propose an asymptotic optimal strategy for the solutions of SDEs.

Example 1 Consider the following two SDEs

$\begin{cases} \textrm{d}X_i(t)=f_i(t,B_t^H,X_i(t))\textrm{d}t+\sigma(t,B_t^H,X_i(t))\textrm{d}B_s^H, \\ X_i(t_0)=x_0^i (i=1,2), \end{cases}$

where $\sigma$, $f_1$, and $f_2$ satisfy the conditions in Theorem 2.3 and $x_0^1\leq{x_0^2}$, $f_1\leq{f_2}$ in $G$.

Set $\tilde{\sigma}=\sigma$ and $h^{(\epsilon)}_i(t,\xi,\omega)\equiv(f_i-\Phi^{(\epsilon)}_t)/\Phi^{(\epsilon)}_\xi (i=1,2),$ we can easily show that

$ h^{(\epsilon)}_1(t,\xi,\omega)\leq{h^{(\epsilon)}_2(t,\xi,\omega)} \forall \epsilon>0. $

Hence, applying Theorem 3.1 and 3.2 to the equations

$\begin{cases} \textrm{d}X^{(\epsilon)}_i(t)=f_i(t,B_t^H,X^{(\epsilon)}_i(t))\textrm{d}t+\sigma^{(\epsilon)}(t,B_t^H,X^{(\epsilon)}_i(t))\textrm{d}B_s^H, \\ X^{(\epsilon)}_i(t_0)=x_0^i (i=1,2), \end{cases}$

we see that $X^{(\epsilon)}_1(t)\leq{X^{(\epsilon)}_2(t)}$ holds for $\forall \epsilon>0$ on the interval where they are both well-defined. Let $\epsilon\rightarrow 0$, by virtue of Theorem 3.3, we obtain that $X_1(t)\leq{X_2(t)}$.

Theorem 4.1 { Let $G\subset[t_0 ,\infty)\times\mathbb{R}^2$, assume $f$, $\sigma$ are finite measurable function on $G$ and satisfy the conditions in Theorem 3.1, then there exists two finite measurable function sequences $f_n$ and $\sigma_n$ such that

$f_n\leq f, \ \sigma_n\leq \sigma, \ \lim\limits_{n\to\infty}f_n=f, \ \ \lim\limits_{n\to\infty}\sigma_n=\sigma.$

Moreover, $\lim\limits_{n\to\infty}X_t^n=X_t $ a.s., where $X_t^n$ and $X_t$ are solutions of equations

$\begin{equation}\label{equa5} X_t^n=x_0+\int_{t_0}^{t}f_n(s,B_s^H,X_s^n)ds+\int_{t_0}^{t}\sigma_n(s,B_s^H,X_s^n)dB_s^H, t\geq t_0 \end{equation}$

(4.1)

and

$\begin{equation}\label{equa6} X_t=x_0+\int_{t_0}^{t}f(s,B_s^H,X_s)ds+\int_{t_0}^{t}\sigma(s,B_s^H,X_s)dB_s^H, t\geq t_0, \end{equation}$

(4.2)

respectively. }

Proof  Since $f$, $\sigma$ are finite measurable functions on $G$, there exists two sequences of simple functions $\{\psi_n\}$ with $|\psi_1|\leq|\psi_2|\leq\cdots\leq|\psi_n|\leq\cdots$, and $\{\varphi_n\}$ with $|\varphi_1|\leq|\varphi_2|\leq\cdots\leq|\varphi_n|\leq\cdots$, such that

$\lim\limits_{n\to\infty}\psi_n=f; \ \ \lim\limits_{n\to\infty}\varphi_n=\sigma.$

Let $f_n=\min\{\psi_n, f\}$, $\sigma_n=\min\{\varphi_n, \sigma\}$, then

$ f_n\leq f, \lim\limits_{n\to\infty}f_n=f; \ \sigma_n\leq \sigma, \lim\limits_{n\to\infty}\sigma_n=\sigma . $

Notice that equations (4.1) and (4.2) are well-defined with regard to the foregoing drift and diffusion coefficients, applying the above elaborated comparison theorems, we have $X_t^n\leq X_t$, and $\lim\limits_{n\to\infty}X_t^n=X_t$.

Remark 4.1 By suitable designing to approximate the drift and diffusion coefficients numerically and using comparison theorems, we could obtain the optimal state of the stochastic dynamical systems.