#### 期刊菜单

Moment Boundedness of Solutions to Nonlinear Stochastic Delay Differential Equations with Markovian Switching and Poisson Jumps

Abstract: This paper investigates the moment boundedness of the solutions to nonlinear stochastic delay differential equations with Markovian switching and Poisson jumps. It is first proved the existence and uniqueness of the solution for such an equation. By using stochastic analysis and inequality techniques, it is then obtained that the solution is moment bounded.

1. 引言

2. 预备知识

${R}^{n}$ 表示n-维的欧式空间。若 $x\in {R}^{n}$$|x|$ 表示欧式范数。记 $R=\left(-\infty ,+\infty \right)$${R}^{+}=\left[0,+\infty \right)$。记 $\left(\Omega ,\mathcal{F},P\right)$ 为一个完备的概率空间，其滤子 ${\mathcal{F}}_{t}$ 满足通常的条件，即 ${\left\{{\mathcal{F}}_{t}\right\}}_{t\ge 0}$ 是右连续的且 ${\mathcal{F}}_{0}$ 包含所有的零测集。令 $\tau >0$$C\left(\left[-\tau ,0\right];{R}^{n}\right)$ 表示所有定义在 $\left[-\tau ,0\right]$ 上的 ${R}^{n}$ -值连续函数 $\phi$ 的全体。 ${C}_{{F}_{0}}^{b}\left(\Omega ;{R}^{n}\right)$ 表示所有 ${\mathcal{F}}_{0}$ -可测的且定义在 $C\left(\left[-\tau ,0\right];{R}^{n}\right)$ 上的有界函数全体，其上的范数为 $‖\phi ‖={\mathrm{sup}}_{-\tau \le \theta \le 0}|\phi \left(\theta \right)|$。对 $\forall x,y\in {R}^{n}$$〈x,y〉$${x}^{\text{T}}y$ 均表示其内积。

${\left\{r\left(t\right)\right\}}_{t\ge 0}$ 代表的是定义在完备概率空间 $\left(\Omega ,\mathcal{F},P\right)$ 上的一个取值为 $S=\left\{1,2,\cdots ,N\right\}$ 的右连续时齐Markov链，其生成元(密度矩阵) $\Gamma ={\left({\gamma }_{ij}\right)}_{N×N}$ 由转移概率矩阵确定，即

$P\left\{r\left(t+\Delta \right)=j|r\left(t\right)=i\right\}=\left\{\begin{array}{ll}{\gamma }_{ij}\Delta +o\left(\Delta \right),\hfill & 当\text{\hspace{0.17em}}i\ne j,\hfill \\ 1+{\gamma }_{ii}\Delta +o\left(\Delta \right),\hfill & 当\text{\hspace{0.17em}}i=j,\hfill \end{array}$

$\begin{array}{l}\text{d}x\left(t\right)=f\left(x\left(t\right),x\left(t-\tau \right),r\left(t\right),t\right)\text{d}t+g\left(x\left(t\right),x\left(t-\tau \right),r\left(t\right),t\right)\text{d}B\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+h\left(x\left(t\right),x\left(t-\tau \right),r\left(t\right),t\right)\text{d}N\left(t\right),\text{\hspace{0.17em}}\text{ }t\ge 0。\end{array}$ (1)

${x}_{0}=\phi =\left\{x\left(t\right):-\tau \le t\le 0\right\}\in {C}_{{F}_{0}}^{b}\left(\Omega ;{R}^{n}\right)$$r\left(0\right)={i}_{0}\in S$(2)

(A1) 对任意的 $h>0$，存在一个常数 ${L}_{h}>0$ 使得对 $\forall \left(i,t\right)\in S×{R}^{+}$${x}_{1},{x}_{2},{y}_{1},{y}_{2}\in {R}^{n}$，有

$\begin{array}{l}|f\left({x}_{1},{y}_{1},i,t\right)-f\left({x}_{2},{y}_{2},i,t\right)|\vee |g\left({x}_{1},{y}_{1},i,t\right)-g\left({x}_{2},{y}_{2},i,t\right)|\\ \vee |h\left({x}_{1},{y}_{1},i,t\right)-h\left({x}_{2},{y}_{2},i,t\right)|\le {L}_{h}\left(|{x}_{1}-{x}_{2}|+|{y}_{1}-{y}_{2}|\right)\end{array}$

${C}^{2,1}\left({R}^{n}×S×{R}^{+};{R}^{+}\right)$ 表示关于变量x二阶连续可导且关于变量t一阶连续可导的全体非负函数 $V\left(x,i,t\right)$ 的集合。给定任意的 $V\left(x,i,t\right)\in {C}^{2,1}\left({R}^{n}×S×{R}^{+};{R}^{+}\right)$，定义算子

$\begin{array}{c}LV\left(x,y,i,t\right)={V}_{t}\left(x,i,t\right)+{V}_{x}\left(x,i,t\right)f\left(x,y,i,t\right)\\ \text{\hspace{0.17em}}+\frac{1}{2}\text{trace}\left[{g}^{\text{T}}\left(x,y,i,t\right){V}_{xx}\left(x,i,t\right)g\left(x,y,i,t\right)\right]\\ \text{\hspace{0.17em}}+\lambda \left[V\left(x+h\left(x,y,i,t\right),i,t\right)-V\left(x,i,t\right)\right]+{\sum }_{j=1}^{N}{\gamma }_{ij}V\left(x,j,t\right)\text{\hspace{0.17em}},\end{array}$

${V}_{x}\left(x,i,t\right)=\left(\frac{\partial V\left(x,i,t\right)}{\partial {x}_{1}},\frac{\partial V\left(x,i,t\right)}{\partial {x}_{2}},\cdots ,\frac{\partial V\left(x,i,t\right)}{\partial {x}_{n}}\right)$

(A2) 令 $H\left(\cdot \right)\in C\left({R}^{n}×\left[-\tau ,\infty \right);{R}^{+}\right)$。假设存在一个函数 $V\in {C}^{2,1}\left({R}^{n}×S×{R}^{+};{R}^{+}\right)$ 和正常数 ${c}_{1},{c}_{2},{c}_{3}$ 满足

${c}_{3}<{c}_{2}$${|x|}^{q}\le V\left(x,i,t\right)\le H\left(x,t\right)$$\forall \left(x,i,t\right)\in {R}^{n}×S×{R}^{+}$

$LV\left(x,y,i,t\right)\le {c}_{1}-{c}_{2}H\left(x,t\right)+{c}_{3}H\left(y,t-\tau \right)$$\forall \left(x,y,i,t\right)\in {R}^{n}×{R}^{n}×S×{R}^{+}$

3. 主要结果

$\begin{array}{l}EV\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\\ =V\left(x\left(0\right),r\left(0\right),0\right)+E{\int }_{0}^{t\wedge {\tau }_{m}}LV\left(x\left(s\right),x\left(s-\tau \right),r\left(s\right),s\right)\text{d}s。\end{array}$

$\begin{array}{l}EV\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\\ \le V\left(x\left(0\right),r\left(0\right),0\right)+{c}_{1}t-{c}_{2}E{\int }_{0}^{t\wedge {\tau }_{m}}H\left(x\left(s\right),s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{c}_{3}E{\int }_{0}^{t\wedge {\tau }_{m}}H\left(x\left(s-\tau \right),s-\tau \right)\text{d}s。\end{array}$

$\begin{array}{c}{\int }_{0}^{t\wedge {\tau }_{m}}H\left(x\left(s-\tau \right),s-\tau \right)\text{d}s={\int }_{-\tau }^{t\wedge {\tau }_{m}-\tau }H\left(x\left(s\right),s\right)\text{d}s\\ \le {\int }_{-\tau }^{0}H\left(x\left(s\right),s\right)\text{d}s+{\int }_{0}^{t\wedge {\tau }_{m}}H\left(x\left(s\right),s\right)\text{d}s。\end{array}$

$\begin{array}{l}EV\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\\ \le V\left(x\left(0\right),r\left(0\right),0\right)+{c}_{1}t+{c}_{3}{\int }_{-\tau }^{0}H\left(x\left(s\right),s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left({c}_{2}-{c}_{3}\right)E{\int }_{0}^{t\wedge {\tau }_{m}}H\left(x\left(s\right),s\right)\text{d}s。\end{array}$

$EV\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\le V\left(x\left(0\right),r\left(0\right),0\right)+{c}_{1}t+{c}_{3}{\int }_{-\tau }^{0}H\left(x\left(s\right),s\right)\text{d}s$

${K}_{1}=V\left(x\left(0\right),r\left(0\right),0\right)+{c}_{3}{\int }_{-\tau }^{0}H\left(x\left(s\right),s\right)\text{d}s$

$E{|x\left(t\wedge {\tau }_{m}\right)|}^{q}\le {K}_{1}+{c}_{1}t$

${m}^{q}P\left({\tau }_{m}\le t\right)\le {K}_{1}+{c}_{1}t$

$m\to \infty$，可得 $P\left({\tau }_{\infty }\le t\right)\to 0$，即 ${\tau }_{\infty }>t$ a.s.由于t的任意性可得

${\tau }_{\infty }=\infty$ a.s.

$\mathrm{sup}{E}_{-\tau \le t<\infty }{|x\left(t\right)|}^{q}<\infty$

$\begin{array}{l}E{\text{e}}^{\epsilon \left(t\wedge {\tau }_{m}\right)}V\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\\ =V\left(x\left(0\right),r\left(0\right),0\right)+E{\int }_{0}^{t\wedge {\tau }_{m}}{\text{e}}^{\epsilon s}LV\left(x\left(s\right),x\left(s-\tau \right),r\left(s\right),s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{t\wedge {\tau }_{m}}\epsilon {\text{e}}^{\epsilon s}V\left(x\left(s\right),r\left(s\right),s\right)\text{d}s,\end{array}$

$\begin{array}{l}E{\text{e}}^{\epsilon \left(t\wedge {\tau }_{m}\right)}V\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\\ \le V\left(x\left(0\right),r\left(0\right),0\right)+E{\int }_{0}^{t\wedge {\tau }_{m}}\epsilon {\text{e}}^{\epsilon s}H\left(x\left(s\right),s\right)\text{d}s+{c}_{1}E{\int }_{0}^{t\wedge {\tau }_{m}}{\text{e}}^{\epsilon s}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-{c}_{2}E{\int }_{0}^{t\wedge {\tau }_{m}}{\text{e}}^{\epsilon s}H\left(x\left(s\right),s\right)\text{d}s+{c}_{3}E{\int }_{0}^{t\wedge {\tau }_{m}}{\text{e}}^{\epsilon \tau }{\text{e}}^{\epsilon \left(s-\tau \right)}H\left(x\left(s-\tau \right),s-\tau \right)\text{d}s。\end{array}$

$\begin{array}{l}E{\text{e}}^{\epsilon \left(t\wedge {\tau }_{m}\right)}V\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\\ \le V\left(x\left(0\right),r\left(0\right),0\right)+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t}+E{\int }_{0}^{t\wedge {\tau }_{m}}\epsilon {\text{e}}^{\epsilon s}H\left(x\left(s\right),s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{c}_{2}E{\int }_{0}^{t\wedge {\tau }_{m}}{\text{e}}^{\epsilon s}H\left(x\left(s\right),s\right)\text{d}s+{c}_{3}{\text{e}}^{\epsilon \tau }E{\int }_{-\tau }^{t\wedge {\tau }_{m}-\tau }{\text{e}}^{\epsilon u}H\left(x\left(u\right),u\right)\text{d}u。\end{array}$

$\begin{array}{l}E{\text{e}}^{\epsilon \left(t\wedge {\tau }_{m}\right)}V\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\\ \le V\left(x\left(0\right),r\left(0\right),0\right)+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t}+{c}_{3}{\text{e}}^{\epsilon \tau }{\int }_{-\tau }^{0}H\left(x\left(s\right),s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{ }\text{ }-\left({c}_{2}-{c}_{3}{\text{e}}^{\epsilon \tau }-\epsilon \right)E{\int }_{0}^{t\wedge {\tau }_{m}}{\text{e}}^{\epsilon s}H\left(x\left(s\right),s\right)\text{d}s。\end{array}$

$\begin{array}{l}E{\text{e}}^{\epsilon \left(t\wedge {\tau }_{m}\right)}V\left(x\left(t\wedge {\tau }_{m}\right),r\left(t\wedge {\tau }_{m}\right),t\wedge {\tau }_{m}\right)\\ \le V\left(x\left(0\right),r\left(0\right),0\right)+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t}+{c}_{3}{\text{e}}^{\epsilon \tau }{\int }_{-\tau }^{0}H\left(x\left(s\right),s\right)\text{d}s。\end{array}$

$m\to \infty$，则

$E{\text{e}}^{\epsilon t}V\left(x\left(t\right),r\left(t\right),t\right)\le V\left(x\left(0\right),r\left(0\right),0\right)+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t}+{c}_{3}{\text{e}}^{\epsilon \tau }{\int }_{-\tau }^{0}H\left(x\left(s\right),s\right)\text{d}s={K}_{2}+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t}$,

$E{|x\left(t\right)|}^{q}\le {K}_{2}{\text{e}}^{-\epsilon t}+\frac{{c}_{1}}{\epsilon }\le {K}_{2}+\frac{{c}_{1}}{\epsilon }<\infty$

NOTES

*通讯作者。

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