Comparison Theorem for Distribution Dependent Neutral SFDEs

TL;DR

This paper establishes existence, uniqueness, and order preservation conditions for distribution dependent neutral stochastic functional differential equations, extending known results and providing new necessary conditions.

Contribution

It introduces new necessary and sufficient conditions for order preservation in distribution dependent neutral SFDEs, extending previous distribution independent results.

Findings

Proved existence and uniqueness of strong solutions.

Established conditions for order preservation.

Extended results to distribution dependent case.

Abstract

In this paper, the existence and uniqueness of strong solutions to distribution dependent neutral SFDEs are proved. We give the conditions such that the order preservation of these equations holds. Moreover, we show these conditions are also necessary when the coefficients are continuous. Under sufficient conditions, the result extends the one in the distribution independent case, and the necessity of these conditions is new even in distribution independent case.

Equations

$$\mathbb{W}_{p}(\mu_{1},\mu_{2})=\inf_{\pi\in\mathbf{C}(\mu_{1},\mu_{2})}\bigg{(}\int_{\mathscr{C}\times\mathscr{C}}\|\xi-\eta\|_{\infty}^{p}\pi(\text{\mbox{d}}\xi,\text{\mbox{d}}\eta)\bigg{)}^{1/p},$$

$$\mbox{d}\big{[}X(t)-D(X_{t})\big{]}=b(t,X_{t},\mathscr{L}_{X_{t}})\mbox{d}t+\sigma(t,X_{t},\mathscr{L}_{X_{t}})\mbox{d}W(t)$$

$$\mbox{d}\big{[}\bar{X}(t)-D(\bar{X}_{t})\big{]}=\bar{b}(t,\bar{X}_{t},\mathscr{L}_{\bar{X}_{t}})\text{\mbox{d}}t+\bar{\sigma}(t,\bar{X}_{t},\mathscr{L}_{\bar{X}_{t}})\mbox{d}W(t),$$

$$\mathbb{E}|D(X_{s,t})|^{2}+\mathbb{E}\|X_{s,t}\|_{\infty}^{2}+\int_{s}^{t}\mathbb{E}\big{\{}|b(r,X_{s,r},\mathscr{L}_{X_{s,r}})|+\|\sigma(r,X_{s,r},\mathscr{L}_{X_{s,r}})\|^{2}\big{\}}\text{\mbox{d}}r<\infty,\ \ t\geq s,$$

$$X_s,(t)-D(X_{s,t})$$

$$+{\int }_s^t\sigma (r,X_{s,r},{\mathcal{L}}_{X_{s,r}})\text{mboxd}W(r),t\ge s.$$

$$\text{mboxd}({\tilde{X}}_{s,}(t)-D({\tilde{X}}_{s,t}))=b(t,{\tilde{X}}_{s,t},{\mathcal{L}}_{{\tilde{X}}_{s,t}}{|}_{\tilde{\mathbb{P}}})\text{mboxd}t+\sigma (t,{\tilde{X}}_{s,t},{\mathcal{L}}_{{\tilde{X}}_{s,t}}{|}_{\tilde{\mathbb{P}}})\text{mboxd}\tilde{W}(t),t\ge s.$$

$$\begin{array}{rl}& |b(t,\xi ,\mu )-b(t,\eta ,\nu ){|}^2+|\bar{b}(t,\xi ,\mu )-\bar{b}(t,\eta ,\nu ){|}^2\\ & \le L(\Vert \xi -\eta {\Vert }_{\infty }^2+{\mathbb{W}}_2(\mu ,\nu {)}^2),t\ge 0,\xi ,\eta \in \mathcal{C};\mu ,\nu \in {\mathcal{P}}_2(\mathcal{C}).\end{array}$$

$$\begin{array}{rl}& \sum _{j=1}^m\Vert {\sigma }_{ij}(t,\xi ,\mu )-{\sigma }_{ij}(t,\eta ,\nu ){\Vert }^2+\Vert {\bar{\sigma }}_{ij}(t,\xi ,\mu )-{\bar{\sigma }}_{ij}(t,\eta ,\nu ){\Vert }^2\\ & \le L|{\xi }^i(0)-D^i(\xi )-{\eta }^i(0)+D^i(\eta ){|}^2,t\ge 0,\xi ,\eta \in \mathcal{C};\mu ,\nu \in {\mathcal{P}}_2(\mathcal{C}),\end{array}$$

$$\begin{array}{r}|b(t,0,{\delta }_0){|}^2+|\bar{b}(t,0,{\delta }_0){|}^2+|\sigma (t,0,{\delta }_0){|}^2+|\bar{\sigma }(t,0,{\delta }_0){|}^2\le \beta (t),t\ge 0,\end{array}$$

$$|D(\xi )-D(\eta )|\le \kappa \underset{1\le i\le n}{\max }\Vert {\xi }^i-{\eta }^i{\Vert }_{\infty }.$$

$$\begin{array}{rl}& |b(t,\xi ,\mu )-b(t,\eta ,\nu ){|}^2+|\bar{b}(t,\xi ,\mu )-\bar{b}(t,\eta ,\nu ){|}^2\\ & \le \alpha (t)(\Vert \xi -\eta {\Vert }_{\infty }^2+{\mathbb{W}}_2(\mu ,\nu {)}^2),t\ge 0,\xi ,\eta \in \mathcal{C};\mu ,\nu \in {\mathcal{P}}_2(\mathcal{C}).\end{array}$$

$$\begin{array}{rl}& \Vert \sigma (t,\xi ,\mu )-\sigma (t,\eta ,\nu ){\Vert }^2+\Vert \bar{\sigma }(t,\xi ,\mu )-\bar{\sigma }(t,\eta ,\nu ){\Vert }^2\\ & \le \alpha (t)(\Vert \xi -\eta {\Vert }_{\infty }^2+{\mathbb{W}}_2(\mu ,\nu {)}^2),t\ge 0,\xi ,\eta \in \mathcal{C};\mu ,\nu \in {\mathcal{P}}_2(\mathcal{C}).\end{array}$$

$$|D(\xi )-D(\eta )|\le \kappa \Vert \xi -\eta {\Vert }_{\infty }.$$

$$X_{s,t}^{(0)}=X_{s,s},{\mu }_{s,t}^{(0)}={\mathcal{L}}_{X_{s,t}^{(0)}},t\ge s.$$

$$\text{mboxd}(X_{s,}^{(n)}(t)-D(X_{s,t}^{(n)}))=b(t,X_{s,t}^{(n)},{\mu }_{s,t}^{(n-1)})\text{mboxd}t+\sigma (t,X_{s,t}^{(n)},{\mu }_{s,t}^{(n-1)})\text{mboxd}W(t),t\ge s,$$

$$\mathbb{E}\underset{t\in [s-r_0,T]}{\sup }|X_{s,}^{(n)}(t){|}^2<\infty ,T>s,n\ge 1.$$

$$\mathbb{E}\underset{t\in [s,s+t_0]}{\sup }|X_{s,}^{(n+1)}(t)-X_{s,}^{(n)}(t){|}^2\le 4{\text{e}}^{-n}\mathbb{E}\underset{t\in [s,s+t_0]}{\sup }|X_{s,}^{(1)}(t){|}^2,n\ge 1.$$

$$\check{b}(t,\xi )=b(t,\xi ,{\mu }_t^{(0)}),\check{\sigma }(t,\xi )=\sigma (t,\xi ,{\mu }_t^{(0)}),t\ge 0,\xi \in \mathcal{C}.$$

$$\text{mboxd}(X^{(1)}(t)-D(X_t^{(1)}))=\check{b}(t,X_t^{(1)})\text{mboxd}t+\check{\sigma }(t,X_t^{(1)})\text{mboxd}W(t),X_0^{(1)}=X_0,t\ge 0.$$

$$|b(t,\xi ,{\mu }_t^{(0)}){|}^2+\Vert \sigma (t,\xi ,{\mu }_t^{(0)}){\Vert }_{HS}^2$$

$$\le 2|b(t,\xi ,{\mu }_t^{(0)})-b(t,0,{\mu }_t^{(0)}){|}^2+2|b(t,0,{\mu }_t^{(0)}){|}^2$$

$$+2\Vert \sigma (t,\xi ,{\mu }_t^{(0)})-\sigma (t,0,{\mu }_t^{(0)}){\Vert }_{HS}^2+2\Vert \sigma (t,0,{\mu }_t^{(0)}){\Vert }_{HS}^2$$

$$\displaystyle\leq H(t)\big{\{}1+\|\xi\|_{\infty}^{2}+\mu_{t}^{(0)}(\|\cdot\|_{\infty}^{2})\big{\}},\ \ \ t\geq 0,\xi\in\mathscr{C}.$$

$$|X^{(1)}(t\wedge {\tau }_N)-D(X_{t\wedge {\tau }_N}^{(1)}){|}^2$$

$$\le 3|X^{(1)}(0)-D(X_0^{(1)}){|}^2+3{\left|{\int }_0^{t\wedge {\tau }_N}\sigma (s,X_s^{(1)},{\mu }_s^{(0)})\text{mboxd}W(s)\right|}^2$$

$$+3{\left|{\int }_0^{t\wedge {\tau }_N}b(s,X_s^{(1)},{\mu }_s^{(0)})\text{mboxd}s\right|}^2.$$

$$|X^{(1)}(t\wedge {\tau }_N){|}^2\le \frac{|X^{(1)}(t\wedge {\tau }_N)-D(X_{t\wedge {\tau }_N}^{(1)})+D(0){|}^2}{1-\kappa }+\frac{|D(X_{t\wedge {\tau }_N}^{(1)})-D(0){|}^2}{\kappa }$$

$$\le \kappa \Vert X_{t\wedge {\tau }_N}^{(1)}{\Vert }_{\infty }^2+c|D(0){|}^2+c\Vert X_0^{(1)}{\Vert }_{\infty }^2+c{\left|{\int }_0^{t\wedge {\tau }_N}\sigma (s,X_s^{(1)},{\mu }_s^{(0)})\text{mboxd}W(s)\right|}^2$$

$$\displaystyle\quad+c\left|\int_{0}^{t\wedge\tau_{N}}b(s,X_{s}^{(1)},\mu_{s}^{(0)})\text{\mbox{d}}s\right|^{2},\ \ t\leq\tau_{N}\ \$$

$$\mathbb{E}\underset{s\in [-r_0,t\wedge {\tau }_N]}{\sup }|X^{(1)}(s){|}^2\le c\mathbb{E}\Vert X_0^{(1)}{\Vert }_{\infty }^2+c|D(0){|}^2$$

$$\displaystyle\qquad\qquad\qquad+H(t)\mathbb{E}\int_{0}^{t\land\tau_{N}}\big{(}1+\|X_{s}^{(1)}\|_{\infty}^{2}+\mu_{s}^{(0)}(\|\cdot\|_{\infty}^{2})\big{)}\text{\mbox{d}}s,\ \ t\geq 0.$$

$$\begin{split}&\mathbb{E}\sup_{s\in[-r_{0},t\land\tau_{N}]}|X^{(1)}(s)|^{2}\leq c\mathbb{E}\|X^{(1)}_{0}\|_{\infty}^{2}+c|D(0)|^{2}\\ &\qquad\qquad\qquad+H(t)\int_{0}^{t}\big{\{}1+\mathbb{E}\sup_{r\in[-r_{0},s\land\tau_{N}]}|X^{(1)}(r)|^{2}+\mu_{s}^{(0)}(\|\cdot\|_{\infty}^{2})\big{\}}\text{\mbox{d}}s,\ \ t\geq 0.\end{split}$$

$$\mathbb{E}\underset{s\in [-r_0,t]}{\sup }|X^{(1)}(s){|}^2<\infty ,t\ge 0.$$

$${\xi }^{(n)}(t)=X^{(n+1)}(t)-X^{(n)}(t),$$

Full text

AMS Subject Classification: 65C30, 65L20

Keywords: Comparison Theorem, Distribution dependent neutral SFDEs, Existence and uniqueness, Order preservation, Wasserstein distance.

1 Introduction

It is well-known that the order preservation is always an important topic in every field of mathematics. In the theory of stochastic processes, the order preservation is called “comparison theorem”. There are order preservations in the distribution sense and in the pathwise sense, the pathwise one implies the distribution one. There are a lot of literature to investigate the comparison theorem. For example: Ikeda and Watanabe [7], O’Brien [10], Skorohod [13] and Yamada [17] for one dimensional stochastic differential equations (SDEs) in the pathwise sense, respectively; Chen and Wang [3] for multidimensional diffusion processes in the distribution sense; Gal’cuk and Davis [4], and Mao [9] for one dimensional SDEs driven by semimartingale in the pathwise sense, to name a few, see also [14, 15]. Moreover, the comparison theorem has been extended to stochastic functional (delay) differential equations (SFDEs), SDEs driven by jumps processes and backward SDEs, we refer reader to see [2, 6, 11, 12, 18, 19], and the references therein.

Recently, in their paper [1], Bai and Jiang made the contribution on the comparison theorem for neutral SFDEs, they give sufficient conditions such that the the comparison theorem holds for this class of stochastic equation. In present paper, we shall study the comparison theorem for distribution dependent neutral SFDEs. Our results cover the ones in [1]. Furthermore, we find the conditions are also necessary.

2 Preliminaries

Throughout the paper, we let $(\mathbb{R}^{n},\langle\cdot,\cdot\rangle,|\cdot|)$ be an $n$-dimensional Euclidean space. Denote $\mathbb{R}^{n\times m}$ by the set of all $n\times m$ matrices endowed with Hilbert-Schmidt norm $\|A\|_{HS}:=\sqrt{\mbox{trace}(A^{\ast}A)}$ for every $A\in\mathbb{R}^{n\times m}$, in which $A^{\ast}$ denotes the transpose of $A$. For fixed $r_{0}>0$, let $\mathscr{C}=C([-r_{0},0];\mathbb{R}^{n})$ denote the family of all continuous functions $h:[-r_{0},0]\rightarrow\mathbb{R}^{n}$, endowed with the uniform norm $\|h\|_{\infty}:=\sup_{-r_{0}\leq\theta\leq 0}|h(\theta)|$. Let $\mathscr{P}(\mathscr{C})$ denote all probability measures on $\mathscr{C}$. For any continuous map $f:[-r_{0},\infty)\rightarrow\mathbb{R}^{n}$ and $t\geq 0$, let $f_{t}\in\mathscr{C}$ be such that $f_{t}(\theta)=f(\theta+t)$ for $\theta\in[-r_{0},0]$. We call $(f_{t})_{t\geq 0}$ the segment of $(f(t))_{t\geq-r_{0}}.$ For $p\geq 2,$ let $\mathscr{P}_{p}(\mathscr{C})$ denote all probability measures on $\mathscr{C}$ with finite $p-$moment, i.e. $\mu(\|\cdot\|^{p}_{\infty})=\int_{\mathscr{C}}\|\xi\|_{\infty}^{p}\mu(\text{\mbox{d}}\xi)<\infty.$ It is well-known that $\mathscr{P}_{p}(\mathscr{C})$ is a polish space under the $L^{p}-$Wasserstein distance

[TABLE]

where $\mathbf{C}(\mu_{1},\mu_{2})$ denotes the class of coupling of $\mu_{1}$ and $\mu_{2}.$ Let $(\Omega,\mathscr{F},\{\mathscr{F}_{t}\}_{t\geq 0},\mathbb{P})$ be a complete filtration probability space, and $\{W(t)\}_{t\geq 0}$ be an $m$-dimensional standard Brownian motion defined on this probability space. For any real numbers $a,b$, we denote $a\vee b=\max\{a,b\},a\wedge b=\min\{a,b\},a^{+}=a\vee 0$ and $a^{-}=-(a\wedge 0).$ $a^{+}(a^{-})$ is called the positive (negative) part of $a.$ For a random variable $Y$ on some probability space $(E,\mathscr{E},\mathbb{P})$, we denote $\mathscr{L}_{Y}|\mathbb{P}$ the distribution of $Y$ under $\mathbb{P}$. In this paper, we consider the following distribution dependent neutral stochastic functional differential equations (NSFDEs) on $\mathbb{R}^{n}$:

[TABLE]

and

[TABLE]

where $D:\mathscr{C}\rightarrow\mathbb{R}^{n}$, which is called neutral term, $b,\bar{b}:[0,\infty)\times\mathscr{C}\times\mathscr{P}(\mathscr{C})\rightarrow\mathbb{R}^{n}$, $\sigma,\bar{\sigma}:[0,\infty)\times\mathscr{C}\times\mathscr{P}(\mathscr{C})\rightarrow\mathbb{R}^{n\times m}$ are measurable, and $\mathscr{L}_{X_{t}}$ denotes the distribution of $X_{t}.$

Definition 2.1**.**

$(1)$ For any $s\geq 0$, a continuous adapted process $(X_{s,t})_{t\geq s}$ on $\mathscr{C}$ is called a (strong) solution of (2.1) from time $s$, if

[TABLE]

and $(X_{s},(t):=X_{s,t}(0))_{t\geq s}$ satisfies $\mathbb{P}$-a.s.

[TABLE]

We say that (2.1) has (strong) existence and uniqueness, if for any $s\geq 0$ and $\mathscr{F}_{s}$-measurable random variable $X_{s,s}$ with $\mathbb{E}\|X_{s,s}\|_{\infty}^{2}<\infty$, the equation from time $s$ has a unique solution $(X_{s,t})_{t\geq s}$. When $s=0$ we simply denote $X_{0,}=X$; i.e. $X_{0,}(t)=X(t),X_{0,t}=X_{t},t\geq 0$.

$(2)$ A couple $(\tilde{X}_{s,t},\tilde{W}(t))_{t\geq s}$ is called a weak solution to (2.1) from time $s$, if $\tilde{W}(t)$ is an $m$-dimensional standard Brownian motion on a complete filtration probability space $(\tilde{\Omega},\{\tilde{\mathscr{F}}_{t}\}_{t\geq s},\tilde{\mathbb{P}})$, and $\tilde{X}_{s,t}$ solves

[TABLE]

$(3)$ (2.1) is said to satisfy weak uniqueness, if for any $s\geq 0$, the distribution of a weak solution $(X_{s,t})_{t\geq s}$ to (2.1) from $s\geq 0$ is uniquely determined by $\mathscr{L}_{X_{s,s}}$.

For future, we need the following assumptions.

1. (A1)

$D(0)=0$ and $D(\xi)\geq D(\eta)$ for $\xi\geq\eta$. 2. (A2)

There exists a constant $L>0$ such that

[TABLE] 3. (A3)

For any $i=1,\ldots,n$,

[TABLE]

here $L$ is in (A2). 4. (A4)

There exists a increasing function $\beta(t)\geq 0$ such that

[TABLE]

where $\delta_{0}$ is the Dirac measure at point $0\in\mathscr{C}.$ 5. (A5)

There exists a $\kappa\in(0,1)$ such that

[TABLE]

3 Existence and Uniqueness

In this section, we investigate the existence and uniqueness of the solution to (2.1). To this end, we use conditions which are weaker than the assumptions above.

1. (A2’)

There exists an increasing function $\alpha:[0,\infty)\rightarrow(0,\infty)$ such that

[TABLE] 2. (A3’)

For $\alpha$ in (A2’),

[TABLE] 3. (A5’)

There exists a $\kappa\in(0,1)$ such that

[TABLE]

Theorem 3.1**.**

Assume (A2’), (A3’), (A4) and (A5’), then the equation (2.1) has s unique strong solution. Moreover, the weak uniqueness holds.

We will prove this result by using the argument of [5] and [16], and we only need to consider the first equation in (2.1). For fixed $s\geq 0$ and $\mathscr{F}_{s}$-measurable $\mathscr{C}$-valued random variable $X_{s,s}$ with $\mathbb{E}\|X_{s,s}\|_{\infty}^{2}<\infty$, we construct the first equation in (2.1) by iterating in distribution as follows. Firstly, let

[TABLE]

For any $n\geq 1,$ let $(X_{s,t}^{(n)})_{t\geq s}$ solve the classical neutral SFDE

[TABLE]

with $X_{s,s}^{(n)}=X_{s,s}$, where $\mu_{s,t}^{(n-1)}:=\mathscr{L}_{X_{s,t}^{(n-1)}}$ and $X^{(n)}_{s,t}(\theta):=X^{(n)}_{s,}(t+\theta)$ for $\theta\in[-r_{0},0]$.

Lemma 3.2**.**

Assume (A2’), (A3’), (A4) and (A5’). Then, for every $n\geq 1$, the neutral SFDE $\eqref{EN}$ has a unique strong solution $X^{(n)}_{s,t}$ with

[TABLE]

Moreover, for any $T>0$, there exists $t_{0}>0$ such that for all $s\in[0,T]$ and $X_{s,s}\in L^{2}(\Omega\rightarrow\mathscr{C};\mathscr{F}_{s})$,

[TABLE]

Proof.

The proof is similar to that of [16, Lemma 2.1] and [5, Lemma 3.2]. Without loss of generality, we may assume that $s=0$ and simply denote $X_{0,}(t)=X(t),X_{0,t}=X_{t},t\geq 0$.

(1) We first prove that the SDE (3.1) has a unique strong solution and (3.2) holds.

For $n=1$, let

[TABLE]

Then (3.1) reduces to

[TABLE]

By (A2’), (A3’), (A4) and (A5’), the coefficients $\check{b}$ and $\check{\sigma}$ satisfy the standard monotonicity condition which implies strong existence, uniqueness and non-explosion for neutral SFDE (3.4), see e.g. [1, Theorem 2.1]. By (A2’), (A3’), (A4) and (A5’), there exists an increasing function $H:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that

[TABLE]

For any $N\in[1,\infty)$ and $\tau_{N}:=\inf\{t\geq 0:|X^{(1)}(t)|\geq N\}$,

[TABLE]

Applying inequality $(x+y)^{2}\leq\frac{x^{2}}{p}+\frac{y^{2}}{1-p}$ for $p\in(0,1)$ and $x,y\geq 0$, we have

[TABLE]

for some constant $c>0$. Noting $\kappa\in(0,1)$, combining this with (A4) and applying the BDG inequality we have

[TABLE]

This implies

[TABLE]

By first applying Gronwall’s Lemma then letting $N\rightarrow\infty$, we arrive at

[TABLE]

Therefore, (3.2) holds for $n=1$.

Now, assuming that the assertion holds for $n=k$ for some $k\geq 1$, we are going to show it for $n=k+1$. Since the proof is similar to repeat the argument above with $(X_{\cdot}^{(k+1)},\mu_{\cdot}^{(k)},X_{\cdot}^{(k)})$ replacing $(X_{\cdot}^{(1)},\mu_{\cdot}^{(0)},X_{\cdot}^{(0)})$, we omit it here.

(2) To prove (3.3), let

[TABLE]

By (A2’) and Itô’s formula, there exists an increasing function $K_{1}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that

[TABLE]

Again using inequality $(x+y)^{2}\leq\frac{x^{2}}{\kappa}+\frac{y^{2}}{1-\kappa}$, we have

[TABLE]

By the BDG inequality and noting $\kappa\in(0,1)$, we obtain

[TABLE]

for some increasing function $K_{2}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}.$

By Gronwall’s Lemma, and since $\mathbb{W}_{2}(\mu_{s}^{(n)},\mu_{s}^{(n-1)})^{2}\leq\mathbb{E}\|\xi_{s}^{(n-1)}\|_{\infty}^{2}$, we obtain

[TABLE]

Taking $t_{0}>0$ such that $t_{0}K_{2}(T)\text{\rm{e}}^{t_{0}K_{2}(T)}\leq\text{\rm{e}}^{-1}$, we arrive at

[TABLE]

Since

[TABLE]

we obtain (3.3). ∎

Proof of Theorem 3.1.

(Existence) For simplicity, we only consider $s=0$ and denote $X_{0,}=X$; i.e. $X_{0,}(t)=X(t),X_{0,t}=X_{t},t\geq 0$.

Let $(X_{t})_{t\in[0,t_{0}]}$ be the unique limit of $(X_{t}^{(n)})_{t\in[0,t_{0}]}$ in Lemma 3.2, then $(X_{t})_{t\in[0,t_{0}]}$ is an adapted continuous process and satisfies

[TABLE]

where $\mu_{t}$ is the distribution of $X_{t}$. Rewriting (3.1), we have

[TABLE]

Then (3.5), (A2’), (A3’), (A5’) and the dominated convergence theorem imply that $\mathbb{P}$-a.s.

[TABLE]

Therefore, $(X_{t})_{t\in[0,t_{0}]}$ solves (2.1) up to time $t_{0}$. Moreover, $\mathbb{E}\sup_{s\in[0,t_{0}]}|X(s)|^{2}<\infty$ follows by (3.5). The same holds for $(X_{s,t})_{t\in[s,(s+t_{0})\land T]}$ and $s\in[0,T]$. So, by solving the equation piecewise in time, and using the arbitrariness of $T>0$, we conclude that (2.1) has a strong solution $(X_{t})_{t\geq 0}$ with

[TABLE]

Uniqueness Let $X$ and $Y$ be two solutions to (2.1), i.e.

[TABLE]

and

[TABLE]

By (A2’), we have

[TABLE]

for an increasing function $\beta_{1}:[0,\infty)\rightarrow[0,\infty)$. Applying inequality $(x+y)^{2}\leq\frac{x^{2}}{\kappa}+\frac{y^{2}}{1-\kappa}$, we have

[TABLE]

Noting that $\mathbb{W}_{2}(\mathscr{L}_{X_{t}},\mathscr{L}_{Y_{t}})^{2}\leq\mathbb{E}\|X_{t}-Y_{t}\|_{\infty}^{2}$, (A3’) and the BDG inequality imply that $\gamma_{t}:=\sup_{s\in[-r_{0},t]}|X(s)-Y(s)|^{2}$ satisfies

[TABLE]

for an increasing function $\beta_{2}:[0,\infty)\rightarrow[0,\infty)$. So, applying Gronwall’s inequality implies

[TABLE]

(Weak uniqueness) Since the proof is similar to that of [16, Theorem 2.1], we omit it here. ∎

4 Comparison Theorem

In order to obtain the comparison theorem for distribution dependent NSFDEs, we introduce the partial order on $\mathscr{C}$. If $x=(x_{1},\cdots,x_{n}),y=(y_{1},\cdots,y_{n})\in\mathbb{R}^{n}$, we call $x\leq y$ if and only if $x_{i}\leq y_{i},i=1,\ldots,n;$ $x<y$ if and only if $x\leq y$ and $x\neq y;$ $x\ll y$ if and only if $x_{i}<y_{i},i=1,\ldots,n.$ For $\xi=(\xi_{1},\cdots,\xi_{n}),\eta=(\eta_{1},\cdots,\eta_{n})\in\mathscr{C},$ we call $\xi\leq\eta$ if and only if $\xi(\theta)\leq\eta(\theta),\theta\in[-r_{0},0];$ $\xi<\eta$ if and only if $\xi\leq\eta$ and $\xi\neq\eta;$ $\xi\ll\eta$ if and only if $\xi(\theta)<\eta(\theta),\theta\in[-r_{0},0];$ for any $\xi,\eta\in\mathscr{C},$ $\xi\wedge\eta$ is defined by $(\xi\wedge\eta)_{i}=\xi_{i}\wedge\eta_{i},i=1,\ldots,n.$ We also define the following partial order associated with the neutral term $D(\cdot),$ that is: $\xi\leq_{D}\eta$ if and only if $\xi\leq\eta$ and $\xi(0)-D(\xi)\leq\eta(0)-D(\eta);$ $\xi<_{D}\eta$ if and only if $\xi\leq_{D}\eta$ and $\xi\neq\eta.$ A function $h$ on $\mathscr{C}$ is called increasing if $h(\xi)\leq h(\eta)$ for $\xi\leq\eta.$ Let $\mu_{1},\mu_{2}\in\mathscr{P}(\mathscr{C}),$ we call $\mu_{1}\leq\mu_{2}$ if and only if $\mu_{1}(h)\leq\mu_{2}(h)$ holds for all increasing function $h\in C_{b}(\mathscr{C})$ which denotes all bounded continuous functions on $\mathscr{C}.$

Denote by $(X(s,\xi;t),\bar{X}(s,\bar{\xi};t))_{t\geq s}$ the solutions to (2.1)-(2.2) with $(X_{s}(s,\xi),\bar{X}_{s}(s,\bar{\xi}))=(\xi,\bar{\xi})$. Let $(X_{t}(s,\xi),\bar{X}_{t}(s,\bar{\xi}))_{t\geq s}$ be the segment process.

Definition 4.1**.**

The distribution dependent NSFDE (2.1)-(2.2) is called D-order-preserving, if for any $s\geq 0$ and $\xi,\bar{\xi}\in L^{2}(\Omega\rightarrow\mathscr{C},\mathscr{F}_{s},\mathbb{P})$ with $\mathbb{P}(\xi\leq_{D}\bar{\xi})=1,$ one has

[TABLE]

Definition 4.2**.**

A function $f:\mathscr{C}\rightarrow\mathbb{R}^{1}$ is called $D$-increasing, if for any $\xi\leq_{D}\eta$, it holds $f(\xi)\leq f(\eta)$. If two probability measures $\mu,\nu$ on $\mathscr{C}$ satisfying $\mu(f)\leq\nu(f)$ for any $D$-increasing function $f$, then we denote $\mu\leq_{D}\nu$.

Remark 4.1**.**

In fact, if $\mu\leq_{D}\nu$, by [8, Theorem 5], there exists $\pi\in\mathbf{C}(\mu,\nu)$ with

[TABLE]

4.1 Sufficient Conditions for Comparison Theorem

In this subsection, we will extend the result in [1] and provide sufficient conditions such that the comparison theorem holds. Due to the difficulty caused by the distribution dependence, the generalization is not trivial.

Theorem 4.2**.**

Let (A1)-(A5)* hold and $b,\bar{b}$ and $\sigma,\bar{\sigma}$ are continuous on $[0,\infty)\times\mathscr{C}\times\mathscr{P}_{2}(\mathscr{C})$. Assume $\xi\leq_{D}\bar{\xi}$ and the following conditions hold.*

1. (i)

The drift terms $b=(b_{1},\ldots,b_{n})$ and $\bar{b}=(\bar{b}_{1},\ldots,\bar{b}_{n})$ are continuous in $t$ and $b_{i}(t,\eta,\mu)\leq\bar{b}_{i}(t,\bar{\eta},\bar{\mu})$ for any $1\leq i\leq n$ provided $\mu,\bar{\mu}\in\mathscr{P}_{2}(\mathscr{C})$ with $\mu\leq_{D}\bar{\mu}$, $\eta,\bar{\eta}\in\mathscr{C}$ with $\eta\leq_{D}\bar{\eta}$ and $\eta^{i}(0)-D^{i}(\eta)=\bar{\eta}^{i}(0)-D^{i}(\bar{\eta})$. 2. (ii)

The diffusion terms $\sigma=(\sigma_{ij})$ and $\bar{\sigma}=(\bar{\sigma}_{ij})$ are continuous in $t$ and $\sigma=\bar{\sigma}.$ Moreover, $\sigma_{ij}(t,\eta,\mu)$ only depends on $t$ and $\eta^{i}(0)-D^{i}(\eta)$.

Then $\mathbb{P}(X_{t}(s,\xi)\leq_{D}\bar{X}_{t}(s,\bar{\xi}),t\geq s)=1.$ Thus, $\mathbb{P}(X_{t}(s,\xi)\leq\bar{X}_{t}(s,\bar{\xi}),t\geq s)=1.$

In the following, for simplicity, let $s=0$, $X(t)=X(s,\xi;t),\bar{X}(t)=\bar{X}(s,\bar{\xi};t),X_{D}(t)=X(t)-D(X_{t}),\bar{X}_{D}(t)=\bar{X}(t)-D(\bar{X}_{t}).$

Define the following stopping times:

[TABLE]

Let $\rho=\min\{\rho_{1},\ldots,\rho_{n}\}$ and $\Upsilon=\min\{\Upsilon_{1},\ldots,\Upsilon_{n}\}$. We firstly give a modified proof of [1, Proposition 3.1] which extends the result there to the case that $D$ is nonlinear.

Proposition 4.3**.**

Assume (A1) and (A5) hold, then we have

[TABLE]

Proof.

Set

[TABLE]

and

[TABLE]

Then it is easy to see that for $i=1,\cdots,n$ and $0\leq t\leq\rho_{i}^{l}$,

[TABLE]

and $\rho=\inf\{\rho^{l}:l\geq 1\}$. Moreover, by the definition of $\rho_{i}$ and $\Upsilon_{i}$, one has

[TABLE]

and

[TABLE]

We only need to prove $\Upsilon\leq\rho^{l}$ for any $l\geq 1$ and $\omega\in\Omega$. To this end, we assume that there exists a $l\geq 1$ and $\omega_{0}\in\Omega$ such that $\rho^{l}(\omega_{0})<\Upsilon(\omega_{0})$. Then there exists a $1\leq n_{0}=n_{0}(\omega_{0})\leq n$ such that $\rho_{n_{0}}^{l}(\omega_{0})=\rho^{l}(\omega_{0})$. Then by (4.5), we have $[X^{n_{0}}_{D}(\rho_{n_{0}}^{l})](\omega_{0})\leq[\bar{X}^{n_{0}}_{D}(\rho_{n_{0}}^{l})](\omega_{0})$. This together with (4.3) implies that $\frac{1}{l}+D^{n_{0}}(\bar{X}_{\rho_{n_{0}}^{l}}(\omega_{0}))-D^{n_{0}}(X_{\rho_{n_{0}}^{l}}(\omega_{0}))\leq 0$. This combining with (4.3) and the monotonicity of $D$ yields

[TABLE]

By (A5), we obtain $\frac{1}{l}-\frac{\kappa}{l}\leq 0$. Since $\kappa\in(0,1)$, this is a contradiction. Thus, we finish the proof. ∎

Remark 4.4**.**

With Proposition 4.3 in hand, repeating the proof of [1, Theorem 3.1], we obtain the following result: If $b,\bar{b}$ and $\sigma,\bar{\sigma}$ do not depend on the distribution, under (A1)-(A5), Theorem 4.2 holds by replacing the condition $b_{i}(t,\eta)\leq\bar{b}_{i}(t,\bar{\eta})$ in (i) with $b_{i}(t,\eta)<\bar{b}_{i}(t,\bar{\eta})$.

Now we intend to prove the distribution dependent case.

Proof of Theorem 4.2.

We first prove the result in Theorem 4.2 holds by replacing the condition $b_{i}(t,\eta,\mu)\leq\bar{b}_{i}(t,\bar{\eta},\bar{\mu})$ in (i) with $b_{i}(t,\eta,\mu)<\bar{b}_{i}(t,\bar{\eta},\bar{\mu})$. For any $n\geq 0,$ let $(X_{s,t}^{(n)})_{t\geq s}$ solve (3.1) with $X_{s,s}^{(n)}=\xi$ and $X_{s,t}^{(0)}=\xi,t\geq s$. Similarly, let

[TABLE]

and $(\bar{X}_{s,t}^{(n)})_{t\geq s}$ solve (3.1) with $\bar{b}$ and $\bar{\sigma}$ in place of $b$ and $\sigma$ and $\bar{X}_{s,s}^{(n)}=\bar{\xi}$. Denote $\bar{\mu}_{s,t}^{(n-1)}:=\mathscr{L}_{\bar{X}_{s,t}^{(n-1)}}$. We should remark that $\{\bar{X}_{s,t}^{(0)}\}_{t\geq s}$ and $\{X_{s,t}^{(0)}\}_{t\geq s}$ are continuous $\mathscr{C}$-valued process. Without loss of generality, we assume $s=0$ and omit the subscript $s$.

[TABLE]

and

[TABLE]

For $n=1$, since $\mu_{t}^{(0)}\leq_{D}\bar{\mu}_{t}^{(0)}$, by (i) and (ii) in Theorem 4.2, we have

1. (1)

$b^{1}$ and $\bar{b}^{1}$ are continuous in $t$ and $b^{1}_{i}(t,\eta)<\bar{b}^{1}_{i}(t,\bar{\eta})$ for any $1\leq i\leq n$ provided $\eta,\bar{\eta}\in\mathscr{C}$ with $\eta\leq_{D}\bar{\eta}$ and $\eta^{i}(0)-D^{i}(\eta)=\bar{\eta}^{i}(0)-D^{i}(\bar{\eta})$. 2. (2)

The diffusion terms $\sigma^{1}=(\sigma^{1}_{ij})$ and $\bar{\sigma}^{1}=(\bar{\sigma}^{1}_{ij})$ are continuous in $t$ and $\sigma^{1}=\bar{\sigma}^{1}.$ Moreover, $\sigma^{1}_{ij}(t,\eta)$ only depends on $t$ and $\eta^{i}(0)-D^{i}(\eta)$.

Then by Remark 4.4, it holds $\mathbb{P}$-a.s.

[TABLE]

Next, assume $\mathbb{P}$-a.s.

[TABLE]

Repeating the proof for $b^{n}$, $\sigma^{n}$, $\bar{b}^{n}$, $\bar{\sigma}^{n}$, $X^{(n-1)}$ in place of $b^{1}$, $\sigma^{1}$, $\bar{b}^{1}$, $\bar{\sigma}^{1}$, $X^{(0)}$, we can prove $\mathbb{P}$-a.s.

[TABLE]

By (3.5), we conclude $\mathbb{P}$-a.s.

[TABLE]

and

[TABLE]

Then the required assertion follows.

In general, if the Assumption (i) in Theorem 4.2 holds, then let $(\bar{b}_{\varepsilon},\bar{X}^{\varepsilon})$ be in Lemma 4.5 below. By the above conclusion, we have $\mathbb{P}$-a.s.

[TABLE]

Letting $\varepsilon$ goes to [math], it follows from Lemma 4.5 below and the continuity of $D$ that $\mathbb{P}$-a.s.

[TABLE]

and

[TABLE]

Thus, we complete the proof. ∎

Lemma 4.5**.**

Let $\bar{b}_{\varepsilon}=\bar{b}+\vec{\varepsilon}$, here $\vec{\varepsilon}=(\varepsilon,\varepsilon,\cdots,\varepsilon)\in\mathbb{R}^{n}$ and $\varepsilon>0$. Let $\bar{X}^{\varepsilon}(t)$ solve (2.2) with $\bar{X}^{\varepsilon}_{0}=\bar{X}_{0}$ and $\bar{b}_{\varepsilon}$ in place of $\bar{b}$. If the conditions in Theorem 4.2 hold, then for any $T>0$, it holds that

[TABLE]

The proof is standard, we omit it here.

4.2 Necessary Conditions for Comparison Theorem

In this subsection, we show the conditions in Theorem 4.2 are also necessary. To this end, we firstly introduce a lemma.

Lemma 4.6**.**

1. (1)

For any $1\leq i\leq n$, $\xi,\eta\in\mathscr{C}$ with $\xi^{i}(0)-D^{i}(\xi)=\eta^{i}(0)-D^{i}(\eta)$, there exists $\zeta\in\mathscr{C}$ such that $\zeta\leq\xi\land\eta$ and $\zeta^{i}(0)-D^{i}(\zeta)=\xi^{i}(0)-D^{i}(\xi)=\eta^{i}(0)-D^{i}(\eta)$. 2. (2)

For $\mu,\nu\in\mathscr{P}_{2}(\mathscr{C})$, there exists $\tilde{\mu}\in\mathscr{P}_{2}(\mathscr{C})$ such that $\tilde{\mu}\leq_{D}\mu$ and $\tilde{\mu}\leq_{D}\nu$.

Proof.

(1) Fix $1\leq i\leq n$, $\xi,\eta\in\mathscr{C}$ with $\xi^{i}(0)-D^{i}(\xi)=\eta^{i}(0)-D^{i}(\eta)$. Without loss of generality, assume $\xi^{i}(0)\leq\eta^{i}(0)$. If $D^{i}(\xi\wedge\eta)=D^{i}(\xi)$, let $\zeta=\xi\wedge\eta$. Otherwise, ${\bf(A1)}$ implies $D^{i}(\xi\wedge\eta)<D^{i}(\xi)$. Let $\mathbf{e}\in\mathscr{C}$ be defined by $\mathbf{e}_{i}(s)=1,s\in[-r_{0},0],1\leq i\leq n$. Define

[TABLE]

By (A5) and $D(0)=0$, we have $h^{i}(r)\geq r(1-\kappa),r\geq 0$ and $h^{i}(r)\leq r(1-\kappa),r\leq 0$. The continuity of $D$ implies that there exists a constant $v>0$ such that

[TABLE]

Let $\zeta=\xi\wedge\eta-v\mathbf{e}$, then it is clear that $\zeta\leq\xi\wedge\eta$. Moreover, it follows from (4.7)

[TABLE]

(2) Fix $\mu,\nu\in\mathscr{P}_{2}(\mathscr{C})$. Let two $\mathscr{C}$-valued random variables $(\Gamma_{1},\Gamma_{2})$ on $(\mathscr{C}^{2},\mathscr{B}(\mathscr{C}^{2}),\mu\times\nu)$ be defined as $\Gamma_{k}(\xi_{1},\xi_{2})=\xi_{k},k=1,2$. Then $\mathscr{L}_{\Gamma_{1}}|\mu\times\nu=\mu$ and $\mathscr{L}_{\Gamma_{2}}|\mu\times\nu=\nu$. Basing on this, we can construct a $\mathscr{C}$-valued random variable $\tilde{\Gamma}$ on $(\mathscr{C}^{2},\mathscr{B}(\mathscr{C}^{2}),\mu\times\nu)$ such that $\tilde{\Gamma}\leq\Gamma_{k}$ and $\tilde{\Gamma}(0)-D(\tilde{\Gamma})\leq\Gamma_{k}(0)-D(\Gamma_{k})$, $k=1,2$. Let $\tilde{\mu}=\mathscr{L}_{\tilde{\Gamma}}|\mu\times\nu$. Then we have $\tilde{\mu}\leq_{D}\mu$ and $\tilde{\mu}\leq_{D}\nu$.

In fact, for any $i=1,\cdots,n$, let $h^{i}$ be defined in (4.6). For any $(\xi_{1},\xi_{2})\in\mathscr{C}^{2}$, let

[TABLE]

then $\alpha\mathbf{e}\leq\xi_{1}\wedge\xi_{2}$. Similarly, let

[TABLE]

then

[TABLE]

Let $\tilde{\alpha}=\alpha\wedge\min_{i}\alpha_{i}$ and $\tilde{\Gamma}(\xi_{1},\xi_{2})=\tilde{\alpha}\mathbf{e},(\xi_{1},\xi_{2})\in\mathscr{C}^{2}$. Thus, we finish the proof. ∎

Theorem 4.7**.**

Let (A1)-(A5)* hold. Assume that $\eqref{neq}$-$\eqref{neq1}$ is D-order-preserving for any complete filtration probability space $(\Omega,\{\mathscr{F}_{t}\}_{t\geq 0},\mathbb{P})$ and $m$-dimensional Brownian motion $W(t)$ thereon. Then for any $1\leq i\leq n$, $\mu,\nu\in\mathscr{P}_{2}(\mathscr{C})$ with $\mu\leq_{D}\nu$, and $\xi,\eta\in\mathscr{C}$ with $\xi\leq_{D}\eta$ and $\xi^{i}(0)-D^{i}(\xi)=\eta^{i}(0)-D^{i}(\eta)$, the following assertions hold:*

1. $(i^{\prime})$

$b_{i}(t,\xi,\mu)\leq\bar{b}_{i}(t,\eta,\nu)$* if $b_{i}$ and $\bar{b}_{i}$ are continuous at points $(t,\xi,\mu)$ and $(t,\eta,\nu)$ respectively.* 2. $(ii^{\prime})$

For any $1\leq j\leq m$, $\sigma_{ij}(t,\xi,\mu)=\bar{\sigma}_{ij}(t,\eta,\nu)$ if $\sigma_{ij}$ and $\bar{\sigma}_{ij}$ are continuous at points $(t,\xi,\mu)$ and $(t,\eta,\nu)$ respectively.

Consequently, when $b,\bar{b}$ and $\sigma,\bar{\sigma}$ are continuous on $[0,\infty)\times\mathscr{C}\times\mathscr{P}_{2}(\mathscr{C})$, conditions $(i)$ with $\mu\leq_{D}\bar{\mu}$ in place of $\mu\leq\bar{\mu}$ and $(ii)$ hold.

We first observe that when $b,\bar{b}$ are continuous on $[0,\infty)\times\mathscr{C}\times\mathscr{P}_{2}(\mathscr{C})$, $(i^{\prime})$ implies $(i)$ with $\mu\leq_{D}\bar{\mu}$ in place of $\mu\leq\bar{\mu}$. Next, we prove when $\sigma,\bar{\sigma}$ are continuous on $[0,\infty)\times\mathscr{C}\times\mathscr{P}_{2}(\mathscr{C})$, $(ii^{\prime})$ implies $(ii)$.

Firstly, taking $\xi=\eta$ and $\mu=\nu$, by the continuity of $\sigma$ and $\bar{\sigma}$, $(ii^{\prime})$ implies $\sigma=\bar{\sigma}$.

Let $\zeta$ and $\tilde{\mu}$ be in Lemma 4.6 associated to $\xi,\eta$ and $\mu,\nu$ and applying $(ii^{\prime})$ twice we obtain

[TABLE]

Since $\sigma=\bar{\sigma}$, this implies $(ii)$.

Now, let $t_{0}\geq 0$, $1\leq i\leq n$, $\mu,\nu\in\mathscr{P}_{2}(\mathscr{C})$ with $\mu\leq_{D}\nu$, and $\xi,\eta\in\mathscr{C}$ with $\xi\leq\eta$ and $\xi^{i}(0)-D^{i}(\xi)=\eta^{i}(0)-D^{i}(\eta)$. To prove $(i^{\prime})$ and $(ii^{\prime})$ for $t=t_{0}$, we construct a family of complete filtration probability spaces $(\Omega,\{\mathscr{F}_{t}^{\varepsilon}\}_{t\geq 0},\mathbb{P}^{\varepsilon})_{\varepsilon\in[0,1)}$, $m$-dimensional Brownian motion $W(t)$, and initial random variables $X_{t_{0}}\leq\bar{X}_{t_{0}}$ as follows.

Firstly, since $\mu\leq_{D}\nu$, by Remark 4.1, we may take $\pi_{0}\in{\mathbf{C}}(\mu,\nu)$ such that

[TABLE]

For any $\varepsilon\in[0,1)$, let

[TABLE]

where $\delta_{(\xi,\eta)}$ is the Dirac measure at point $(\xi,\eta)$. Let $\mathbb{P}_{0}$ be the standard Wiener measure on $\Omega_{0}:=C([0,\infty)\rightarrow\mathbb{R}^{m})$, and let $\mathscr{F}_{0,t}$ be the completion of $\sigma(\omega_{0}\mapsto\omega_{0}(s):s\leq t)$ with respect to the Wiener measure. Then the coordinate process $\{W_{0}(t)\}(\omega_{0}):=\omega_{0}(t),\ \omega_{0}\in\Omega_{0},t\geq 0$ is an $m$-dimensional Brwonian motion on the filtered probability space $(\Omega_{0},\{\mathscr{F}_{t}^{0}\}_{t\geq 0},\mathbb{P}_{0})$.

Next, for any $\varepsilon\geq 0$, let $\Omega=\Omega_{0}\times\mathscr{C}^{2},\ \mathbb{P}^{\varepsilon}=\mathbb{P}_{0}\times\pi_{\varepsilon}$ and $\mathscr{F}_{t}^{\varepsilon}$ be the completion of $\mathscr{F}_{0,t}\times\mathscr{B}(\mathscr{C}^{2})$ under the probability measure $\mathbb{P}^{\varepsilon}$. Then the process

[TABLE]

is an $m$-dimensional Brownian motion on the complete probability space $(\Omega,\{\mathscr{F}_{t}^{\varepsilon}\}_{t\geq 0},\mathbb{P}^{\varepsilon})$.

Finally, let

[TABLE]

They are $\mathscr{F}_{t_{0}}^{\varepsilon}$-measurable random variables with

[TABLE]

By $\xi\leq\eta$ with $\xi^{i}(0)-D^{i}(\xi)=\eta^{i}(0)-D^{i}(\eta)$ and (4.8), (4.9), we have

[TABLE]

So, letting $(X_{t},\bar{X}_{t})_{t\geq t_{0}}$ be the segment process of the solution to (2.1) and (2.2) with initial value $(X_{t_{0}},\bar{X}_{t_{0}})$, the D-order preservation implies

[TABLE]

Let $\mathbb{E}^{\varepsilon}$ be the expectation for $\mathbb{P}^{\varepsilon}$. With the above preparations, we are able to prove $(i^{\prime})$ and $(ii^{\prime})$ as follows.

Proof of $(i^{\prime})$.

Let $b_{i},\bar{b}_{i}$ be continuous at points $(t_{0},\xi,\mu)$ and $(t_{0},\eta,\nu)$ respectively. We intend to prove $b_{i}(t_{0},\xi,\mu)\leq b_{i}(t_{0},\eta,\nu)$. Otherwise, there exists a constant $c_{0}>0$ such that

[TABLE]

Let $\mu_{\varepsilon},\nu_{\varepsilon}$ be in (4.10). Obviously, $\{\mu_{\varepsilon},\nu_{\varepsilon}\}_{\varepsilon\in[0,1)}$ are bounded in $\mathscr{P}_{2}(\mathscr{C})$ and, as $\varepsilon\rightarrow 0$, $\mu_{\varepsilon}\rightarrow\mu,\ \nu_{\varepsilon}\rightarrow\nu$ weakly. Consequently,

[TABLE]

Combining this with (4.13) and the continuity of $b$ and $\bar{b}$, there exists $\varepsilon\in(0,1)$ such that

[TABLE]

Now, consider the event

[TABLE]

Then

[TABLE]

By (2.1), (2.2) and (4.12), for any $s\geq 0$, $\mathbb{P}^{\varepsilon}$-a.s.

[TABLE]

By (A2) and the non-explosion of the solution to (2.1) and (2.2), taking conditional expectation in (4.17) with respect to $\mathscr{F}^{\varepsilon}_{t_{0}}$, we obtain $\mathbb{P}^{\varepsilon}$-a.s.

[TABLE]

By (4.15), this implies

[TABLE]

Combining this with the fact that $b_{i}$ and $\bar{b}_{i}$ are continuous at points $(t_{0},\xi,\mu)$ and $(t_{0},\eta,\nu)$ respectively, and using (A2), the non-explosion and continuity of the solution to (2.1) and (2.2), taking $s\downarrow 0$ we derive $\mathbb{P}^{\varepsilon}$-a.s.

[TABLE]

This together with (4.15) and (4.10) leads to $\mathbb{P}^{\varepsilon}$-a.s.

[TABLE]

which is impossible according to (4.14) and (4.16). Therefore, $b_{i}(t_{0},\xi,\mu)\leq\bar{b}_{i}(t_{0},\eta,\nu)$ has to be true. ∎

Proof of $(ii^{\prime})$.

Let $\sigma_{ij}$ and $\bar{\sigma}_{ij}$ be continuous at points $(t_{0},\xi,\mu)$ and $(t_{0},\eta,\nu)$ respectively. If $\sigma_{ij}(t_{0},\xi,\mu)\neq\bar{\sigma}_{ij}(t_{0},\eta,\nu)$, by (A3’), there exist constants $c_{1}>0$ and $\varepsilon\in(0,1)$ such that

[TABLE]

For any $n,l\geq 1$, let

[TABLE]

Let $g_{n}(s)=\text{\rm{e}}^{ns}-1.$ Then $g_{n}\in C_{b}^{2}((-\infty,0])$. By the D-order preservation we have $X^{i}_{t}\leq_{D}\bar{X}^{i}_{t},\ t\geq t_{0}$. So, Letting $Z^{i}(s)=(X^{i}-\bar{X}^{i})(s)-D^{i}(X_{s})+D^{i}(\bar{X}_{s})$ and applying Itô’s formula, we obtain $\mathbb{P}^{\varepsilon}$-a.s.

[TABLE]

By (4.15) and $\xi^{i}(0)-D^{i}(\xi)=\eta^{i}(0)-D^{i}(\eta),$ this implies

[TABLE]

for all $n,l\geq 1$ and $t>t_{0}$.

Take $l\geq 2|b_{i}(t_{0},\xi,\mu_{\varepsilon})-\bar{b}^{i}(t_{0},\eta,\nu_{\varepsilon})|$ and $n>\frac{2\text{\rm{e}}l}{c_{1}}$, we obtain

[TABLE]

But by ${\bf(A3)}$, (4.18) and the continuity of the solution, on the set $A$ we have

[TABLE]

So, (4.19) implies $\mathbb{P}^{\varepsilon}(A)=0$, which contradicts (4.16). Hence,

[TABLE]

∎