\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 285, pp. 1--23.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/285\hfil Exponential $P$-stability] {Exponential $P$-stability of stochastic $\nabla$-dynamic equations on disconnected sets} \author[H. D. Nguyen, T. D. Nguyen, A. T. Le \hfil EJDE-2015/285\hfilneg] {Huu Du Nguyen, Thanh Dieu Nguyen, Anh Tuan Le} \address{Huu Du Nguyen \newline Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam} \email{dunh@vnu.edu.vn} \address{Thanh Dieu Nguyen \newline Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam} \email{dieunguyen2008@gmail.com} \address{Anh Tuan Le \newline Faculty of Fundamental Science, Hanoi University of Industry, Tu Liem district, Ha Noi, Vietnam} \email{tuansl83@yahoo.com} \thanks{Submitted April 4, 2013. Published November 11, 2015.} \subjclass[2010]{60H10, 34A40, 34D20, 39A13, 34N05} \keywords{Differential operator; dynamic equation; exponential stability; \hfill\break\indent It\^{o}'s formula; Lyapunov function} \begin{abstract} The aim of this article is to consider the existence of solutions, finiteness of moments, and exponential $p$-stability of stochastic $\nabla$-dynamic equations on an arbitrary closed subset of $\mathbb{R}$, via Lyapunov functions. This work can be considered as a unification and generalization of works dealing with random difference and stochastic differential equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} The direct method has become the most widely used tool for studying the exponential stability of stochastic equations. For differential equations, we mention the very interesting book by Khas'minskii \cite{Has} in which author uses the Lyapunov functions to study stability. Foss and Konstantopoulos \cite{FK} presented an overview of stochastic stability methods, mostly motivated by stochastic network applications. Socha \cite {So} considered the exponential $p$-stability of singularly perturbed stochastic systems for the ``slow" and ``fast" components of the full-order system. Govindan \cite{Go} proved the existence and uniqueness of a mild solution under two sets of hypotheses and considered the exponential second moment stability of the solution process for stochastic semilinear functional differential equations in a Hilbert space. We also refer to \cite{Mao,Mao1} in which authors considered stochastic asymptotic stability and boundedness for stochastic differential equations with respect to semimartingale via multiple Lyapunov functions. The long-time behavior of densities of the solutions is studied in \cite{Ru} by using Khas'minskii function. For random difference systems, we can refer the reader to \cite{Pa,Sh,Sh1}, for stability of nonlinear systems. Recently, a method for the unified analysis of equations of motion in continuous and discrete cases within the framework of the theory of time scales has drawn a lot attention. For deterministic cases, in \cite{Da}, author used the Lyapunov function of quadratic form to study the stability of linear dynamic equations. Hoffacker and Tisdell examined the stability and instability of the equilibrium point of nonlinear dynamic equations \cite{HT}. Martynyuk presented systematically the stability theory of dynamic equations in \cite{Mar}. While the stability of deterministic dynamic equations on time scales has been investigated for a long time, as far as we know, there is not much in mathematical literature for the stochastic case, and no work dealing with the stability of stochastic dynamic equations. Here, we mention some of the first attempts on this direction. In \cite{GS}, the authors developed the theory of Brownian motion. Sanyal in his Ph. D. Dissertation \cite{San} tries to define \emph{stochastic integral and stochastic dynamic equations} on time scale with the positive graininess. Lungan and Lupulescu in \cite{LL} consider random dynamical systems with random $\Delta$-integral. Gravagne and Robert deal with the bilateral Laplace transforms in \cite{GR}. The Doob-Meyer decomposition theorem and definition of stochastic $\nabla$-integral with respect to square integrable martingale on any arbitrary time scale also It\^{o}'s formula are studied in \cite{{NHDu1},{NHDu2}}. Recently, Bohner et al \cite{Bo} investigate stochastic dynamic equations on time scale by considering an integral with respect to the restriction of a standard Wiener process on time scale. However, this way can not be applied to define the stochastic integral in general case since when one deals with a martingale defined on time scale and we do not know whenever it can be extended to a regular martingale on $\mathbb{R}$. The aim of this article is to use Lyapunov functions to consider the existence, finiteness of moments, and long term behavior of solutions for $\nabla$-stochastic dynamic equations on arbitrary closed subset of $\mathbb{R}$. We study \begin{gather*} d^\nabla X(t)=f(t, X(t_-))d^\nabla t+g(t,X(t_-))d^\nabla M(t)\\ X(a)=x_a\in\mathbb{R}^d, \quad t\in\mathbb{T}_a, \end{gather*} where $(M_t)_{t\in \mathbb{T}_a}$ is a $\mathbb{R}$-valued square integrable martingale and $f: \mathbb{T}_a\times\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{T}_a\times\mathbb{R}\to\mathbb{R}$ are two Borel functions. We emphasis that martingale $M$ is defined only on $\mathbb{T}_a$. This work can be considered as a unification and generalization of works dealing with these areas of stochastic difference and differential equations. In working on stochastic multi-dimensional dynamic equations with respect to discontinuous martingale on time scales, it rises many difficulties, especially the complicated calculations and they require some improvements. Besides, some estimates of stochastic calculus for continuous time are not automatically valid on arbitrary time scale and we need to change them into a suitable form to obtain similar results. The organization of this paper is as follows. We introduce some basic notion and definitions for time scale and for square integrable martingales in Section \ref{sec2}. Section \ref{S1} deals with the existence and the finiteness of moments of solutions for stochastic dynamic equations with respect to a square integrable martingale in case the coefficients satisfy locally Lipschitz conditions. Section \ref{S2} is concerned with necessary and sufficient conditions for the exponential $p$-stability of stochastic dynamic equations. \section{Preliminaries} \label{sec2} Let $\mathbb{T}$ be a closed subset of $\mathbb{R}$, enclosed with the topology inherited from the standard topology on $\mathbb{R}$. Let $\sigma(t)=\inf\{s\in\mathbb{T}: s>t\}, \mu(t)=\sigma(t)-t$ and $\rho(t)=\sup\{s\in\mathbb{T}: st$, {\it left-dense} if $\rho(t)=t$, {\it left-scattered} if $\rho(t)t_0. \end{equation} Also if $p\in\mathcal{R}$, $e_{\ominus p}(t, t_0)$ is the solution of the equation \begin{equation*} y^\nabla(t) =-p(t_-)y(t),\quad y(t_0)=1, \quad t>t_0, \end{equation*} where $\ominus p(t)=\frac{-p(t)}{1+\mu(t)p(t)}$. Later, we need the following lemma. \begin{lemma}[\cite{Pet,NHDu2}] \label{lem1.1} Let $u(t)$ be a regulated function and $ u_ a, \alpha\in\mathbb{R}_+$. Then, the inequality $$ u(t)\leq u_a+\alpha\int_{a}^t u(\tau_-)\nabla \tau\quad \forall t\in\mathbb{T}_a $$ implies $$ u(t)\leq u_a{e}_{\alpha}(t,a)\quad \forall t\in\mathbb{T}_a. $$ \end{lemma} Let $(\Omega, \mathcal{F},\{\mathcal{F}_t\}_{t\in\mathbb{T}_a}, \mathbb{P})$ be a probability space with filtration $\{\mathcal{F}_t\}_{t\in\mathbb{T}_a}$ satisfying the usual conditions (i.e., $\{\mathcal{F}_t\}_{t\in\mathbb{T}_a}$ is increasing and $\cap\{ \mathcal{F}_{\rho(s)}:s\in \mathbb{T}, s>t\}=\mathcal{F}_t$ for all $t\in \mathbb{T}_a$ while $\mathcal{F}_a$ contains all $P$-null sets). Denote by $\mathcal{M}_2$ the set of the square integrable $\mathcal{F}_t$-martingales and by $\mathcal{M}_2^r$ the subspace of the space $\mathcal{M}_2$ consisting of martingales with continuous characteristics. For any $M\in\mathcal{M}_2$, set \begin{equation*} \widehat{M}_t=M_t-\sum_{s\in (a, t]}(M_s-M_{\rho(s)}). \end{equation*} It is clear that $\widehat M_t $ is an $\mathcal{F}_t$-martingale and $\widehat{M}_t=\widehat{M}_{\rho(t)}$ for any $t\in\mathbb{T}$. Further, \begin{equation}\label{e1.2} \langle \widehat M\rangle_t=\langle M\rangle_t-\sum_{ s\in (a, t]} (\langle M\rangle_s-\langle M\rangle_{\rho(s)}). \end{equation} Therefore, $M\in\mathcal{M}_2^r$ if and only if $\widehat{M}\in\mathcal{M}_2^r$. In this case, $\widehat{M}$ can be extended to a regular martingale defined on $\mathbb{R}$. Denote by $\mathfrak{B}$ the class of Borel sets in $\mathbb{R}$ whose closure does not contain the point $0$. Let $\delta(t,A)$ be the number of jumps of the $M$ on the $(a,t]$ whose values fall into the set $ A\in\mathfrak{B}$. Since the sample functions of the martingale $M$ are cadlag, the process $\delta(t, A)$ is defined with probability $1$ for all $t\in\mathbb{T}_a, A\in\mathfrak{B}$. We extend its definition over the whole $\Omega$ by setting $\delta(t, A)\equiv 0$ if the sample $t\to M_t(\omega)$ is not cadlag. Clearly the process $\delta(t, A)$ is $\mathcal{F}_t$-adapted and its sample functions are nonnegative, monotonically nondecreasing, continuous from the right and take on integer values. We also define $\widehat{\delta}(t,A)$ for $\widehat{M}_t$ by a similar way. Let $\widetilde \delta(t,A)=\sharp \{s\in (a, t]: M_s-M_{\rho(s)}\in A\}$. It is evident that \begin{equation}\label{e1.3b} \delta(t,A)=\widehat \delta(t,A)+\widetilde \delta(t,A). \end{equation} Further, for fixed $t$, $\delta(t,\cdot),\widehat \delta(t,\cdot)$ and $\widetilde \delta(t,\cdot)$ are measures. The functions $\delta(t, A), \widehat \delta(t,A)$ and $\widetilde \delta(t,A), t\in\mathbb{T}_a$ are $\mathcal{F}_t$-regular submartingales for fixed $A$. By Doob-Meyer decomposition, each process has a unique representation of the form \begin{gather*} \delta (t, A)=\zeta(t, A)+\pi(t, A),\quad \widehat\delta (t, A)=\widehat\zeta(t, A)+\widehat\pi(t, A),\\ \widetilde\delta (t, A)=\widetilde\zeta(t, A)+\widetilde\pi(t, A), \end{gather*} where $\pi(t, A), \widehat\pi(t, A)$ and $ \widetilde\pi(t, A)$ are natural increasing integrable processes and $\zeta(t, A), \widehat\zeta(t, A)$, $\widetilde\zeta(t, A)$ are martingales. We find a version of these processes such that they are measures when $t$ is fixed. By denoting \begin{equation*} \widehat{M}^c_t=\widehat{M}_t-\widehat{M}^d_t, \end{equation*} where \begin{equation*} \widehat{M}_t^d=\int_a^t\int_{\mathbb{R}}u\widehat\zeta(\nabla \tau, du), \end{equation*} we obtain \begin{equation}\label{e1.3} \langle \widehat{M}\rangle_t =\langle \widehat{M}^c\rangle_t+\langle \widehat{M}^d\rangle_t, \quad \langle \widehat{M}^d\rangle_t =\int_a^t\int_{\mathbb{R}}u^2\widehat\pi(\nabla \tau, du). \end{equation} Throughout this article, we suppose that $\langle M\rangle_t$ is absolutely continuous with respect to Lebesgue measure $\mu_\nabla$, i.e., there exists $\mathcal{F}_t$-adapted progressively measurable process $K_t$ such that \begin{equation}\label{e1.4} \langle M\rangle_t=\int_{a}^tK_\tau\nabla \tau. \end{equation} Further, for any $T\in \mathbb{T}_a$, \begin{equation}\label{e1.5} \mathbb{P}\{\sup_{a\leq t\leq T}|K_t|\leq N\}=1, \end{equation} where $N$ is a constant (possibly depending on $T$). The relations \eqref{e1.2}, \eqref{e1.3} imply that $\langle\widehat{M}^c\rangle_t$ and $\langle \widehat{M}^d\rangle_t$ are is absolutely continuous with respect to $\mu_\nabla$ on $\mathbb{T}$. Thus, there exists $\mathcal{F}_t$-adapted, progressively measurable bounded process $\widehat{K}^c_t$ and $\widehat{K}^d_t$ satisfying \begin{equation*} \langle\widehat{M}^c\rangle_t=\int_a^t\widehat{K}^c_\tau\nabla\tau,\quad \langle\widehat{M}^d\rangle_t=\int_a^t\widehat{K}^d_\tau\nabla\tau, \end{equation*} and the following relation holds \begin{equation*} \mathbb{P}\{\sup_{a\leq t\leq T}\widehat{K}_t^c+\widehat{K}^d_t\leq {N}\}=1. \end{equation*} Moreover, it is easy to show that $\widehat{\pi}(t, A)$ is absolutely continuous with respect to $\mu_\nabla$ on $\mathbb{T}$, that is, it can be expressed as \begin{equation}\label{e1.6} \widehat{\pi}(t, A)=\int_a^t\widehat{\Upsilon}(\tau, A)\nabla\tau, \end{equation} with an $\mathcal{F}_t$-adapted, progressively measurable process $\widehat{\Upsilon}(t, A)$. Since $\mathfrak{B}$ is generated by a countable family of Borel sets, we can find a version of $\widehat{\Upsilon}(t, A)$ such that the map $t\to \widehat{\Upsilon}(t, A)$ is measurable and for $t$ fixed, $\widehat{\Upsilon}(t, \cdot)$ is a measure. Hence, from \eqref{e1.3} we see that $$ \langle\widehat{M}^d\rangle_t =\int_a^t\int_{\mathbb{R}}u^2\widehat{\Upsilon}(\tau, du))\nabla\tau. $$ This means that $$ \widehat{K}^d_t=\int_{\mathbb{R}}u^2\widehat{\Upsilon}(t, du)). $$ For the process $\widetilde\pi(t, A)$ we can write \begin{equation*} \widetilde\pi(t, A) =\sum_{s\in (a, t]}\mathbb{E} [1_A(M_s-M_{\rho(s)})\big |\mathcal{F}_{\rho(s)}]. \end{equation*} Putting \[ \widetilde{\Upsilon}(t, A) =\begin{cases} \frac{\mathbb{E} [1_A(M_t-M_{\rho(t)}) |\mathcal{F}_{\rho(t)}]}{\nu(t)} &\text{if }\nu(t)>0,\\ 0 &\text{if } \nu(t)=0 \end{cases} \] yields \begin{equation}\label{e1.7} \widetilde\pi(t, A)=\int_a^t \widetilde {\Upsilon}(\tau, A)\nabla \tau. \end{equation} Further, by the definition if $\nu(t)>0$ we have \begin{equation}\label{bs} \int_\mathbb{R} u{\widetilde\Upsilon}(t, du)=\frac{ \mathbb{E} [M_t-M_{\rho(t)} \big|\mathcal{F}_{\rho(t)}]}{\nu(t)}=0, \end{equation} and $$ \int_\mathbb{R} u^2{\widetilde\Upsilon}(t, du)=\frac{ \mathbb{E} [(M_t-M_{\rho(t)})^2 \big|\mathcal{F}_{\rho(t)}]}{\nu(t)} =\frac{\langle M\rangle_t-\langle M\rangle_{\rho(t)}}{\nu(t)}. $$ Let ${\Upsilon}(t, A)=\widehat {\Upsilon}(t, A)+\widetilde{\Upsilon}(t, A)$. From \eqref{e1.3b} we see that $$ \pi(t, A)=\int_a^t{\Upsilon}(\tau, A)\nabla\tau. $$ Denote by $\mathcal{L}_1^{\rm loc}(\mathbb{T}_a, \mathbb{R})$ the family of real valued, $\mathcal{F}_t$-progressively measurable processes $f(t)$ with $\int_a^T|f(\tau)|\nabla\tau<+\infty$\; a.s. for every $T>a$ and by $\mathcal{L}_2(\mathbb{T}_a; M)$ the space of all real valued, $\mathcal{F}_t$-predictable processes $\phi(t)$ satisfying $\mathbb{E}\int_a^T\phi^2(\tau)\nabla\langle M\rangle_\tau<\infty$, for any $T>a$. Consider a $d$-tuple of semimartingales $X(t) = ( X _1(t) , \dots, X_d(t))$ defined by $$ X_i(t)=X_i(a)+\int_a^t f_i(\tau)\,\nabla \tau + \int_a^t g_i(\tau)\nabla M_\tau, $$ where $f_i\in \mathcal{L}_1^{\rm loc}(\mathbb{T}_a, \mathbb{R})$ and $g_i\in\mathcal{L}_2(\mathbb{T}_a;M)$ for $i=\overline{1,d}$. For any twice differentiable function $V$, put \begin{equation}\label{e1.8} \begin{aligned} &\mathcal{A}V(t,x)\\ &=\sum_{i=1}^d \frac{\partial V(t,x)}{\partial x_i}(1-1_{\mathbb{I}}(t))f_i(t) +\Big(V(t, x+f(t)\nu(t))-V(t,x)\Big)\Phi(t)\\ &\quad +\frac{1}{2}\sum_{i,j} \frac{\partial^2 V(t,x)}{\partial x_ix_j}g_i(t)g_j(t) \widehat{K}^c_t - \sum_{i=1}^d \frac{\partial V(t,x)}{\partial x_i}g_i(t) \int_\mathbb{R} u\widehat{\Upsilon}(t, du)\\ &\quad +\int_{\mathbb{R}}(V\big(t,x+f(t)\nu(t)+g(t)u\big) -V(t, x+f(t)\nu(t))){\Upsilon}(t, du), \end{aligned} \end{equation} with $f=(f_1, f_2,\dots, f_d)$; $ g=(g_1, g_2,\dots, g_d)$ and $$ \Phi(t)=\begin{cases} 0&\text{if $t$ is left-dense}\\ 1/\nu(t) &\text{if $t$ is left-scattered}. \end{cases} $$ Let $C^{1,2}(\mathbb{T}_a\times \mathbb{R}^d; \mathbb{R})$ be the set of all functions $V(t, x)$ defined on $\mathbb{T}_a\times \mathbb{R}^d$, having continuous $\nabla$-derivative in $t$ and continuous second derivative in $x$. Using the It\^{o}'s formula in \cite{NHDu2} we see that for any $V\in C^{1,2}(\mathbb{T}_a\times \mathbb{R}^d; \mathbb{R}_+)$ \begin{equation}\label{e1.9} V(t,X(t))-V(a, X(a)) -\int_a^t\Big(V^{\nabla _\tau}(\tau,X({\tau_-}))+\mathcal{A}V(\tau,X({\tau_-})) \Big)\nabla\tau \end{equation} is a locally integrable martingale, where $V^{\nabla _t}$ is partial $\nabla$-derivative of $V(t,x)$ in $t$. \section{Existence of solutions and finiteness of moments for stochastic dynamic equations} \label{S1} Consider a {$\nabla$-stochastic dynamic equations on $\mathbb{T}$} of the form \begin{equation}\label{e1.10} \begin{gathered} d^\nabla X(t)=f(t, X(t_-))d^\nabla t+g(t,X(t_-))d^\nabla M(t) \\ X(a)=x_a\in\mathbb{R}^d, \quad t\in\mathbb{T}_a, \end{gathered} \end{equation} where $f: \mathbb{T}_a\times\mathbb{R}^d\to\mathbb{R}^d$ and $g:\mathbb{T}_a\times\mathbb{R}^d\to\mathbb{R}^d$ are two Borel functions. Under the global Lipschitz and linear growth rate conditions of the coefficients $f, g$, there exists uniquely a solution for Cauchy problem \eqref{e1.10} (see: \cite{NHDu2}). We now consider the case where the coefficients are locally Lipschitz. \begin{theorem}\label{thm1.2} Suppose that for any $k>0$ and $T>a$, there exists a constant $L_{T,k}>0$ such that \begin{equation}\label{e1.11} \|f(t,x)-f(t,y)\|^2\vee \|g(t,x)-g(t,y)\|^2\leq L_{T,k}\|x-y\|^2, \end{equation} for all $x,y\in\mathbb{R}^d$ with $\|x\|\vee\|y\|\leq k$ and $t\in[a, T]$. Further, there are positive constants $c=c(T); b=b(T)$ and a nonnegative function $V\in C^{1,2}([a, T]\times \mathbb{R}^d; \mathbb{R}_+)$ satisfying \begin{equation}\label{e1.12} V^{\nabla_t}(t,x)+\mathcal{A}V(t,x)\leq {c}V(t_-,x)+b \quad \forall (t,x)\in[a, T]\times\mathbb{R}^d, \end{equation} and $\lim_{x\to\infty}\inf_{t\in[a, T]} V(t,x)=\infty$. Then, \eqref{e1.10} has a unique solution $X_{a,x_a}(t)$ defined on $\mathbb{T}_a$. In addition, if there exists a positive constant $c_1=c_1(T)$ such that \begin{equation}\label{e1.13} c_1\|x\|^p\leq V(t, x)\quad \forall (t,x)\in [a, T]\times\mathbb{R}^d, \end{equation} then $$ \mathbb{E} \|X_{a,x_a}(t)\|^p\leq \frac{1}{c_1}(V(a,x_a) +\frac{b}{c})e_{c}(t,a)\quad\forall t\in [a, T]. $$ \end{theorem} \begin{proof} For each $k\geq k_0=[\|x_a\|]+1$, define the truncation function $$ f_k(t,x)=\begin{cases} f(t,x) &\text{if }\|x\|\leq k\\ f(t, \frac{kx}{\|x\|}) &\text{if }\|x\|> k, \end{cases} $$ and $g_k(t,x)$ is defined by a similar way. The functions $f_k$ and $g_k$ satisfy the global Lipschitz condition and the linear growth rate condition. Hence, by \cite[Theorem 3.2]{NHDu2} there exists a unique solution $X_k(\cdot)$ to the equation \begin{equation}\label{e1.14} \begin{gathered} d^\nabla X(t)=f_k(t, X(t_-))d^\nabla t+g_k(t,X(t_-))d^\nabla M(t)\\ X(a)=x_a\in\mathbb{R}^d, \;\forall t\in[a, T]. \end{gathered} \end{equation} Define the stopping time $$ \theta_k= \inf\{t\in[a, T]: |X_k(t)|\geq k\}, \quad \theta_{k_0}=a. $$ It is easy to see that $\theta_k$ is increasing and \begin{equation}\label{e1.15} X_k(t)=X_{k+1}(t)\quad \text{if } a\leq t\leq \theta_k. \end{equation} Let $\theta_\infty=\lim_{k\to\infty}\theta_k$ and the process $X_{a,x_a}(t)=X(t)$, $a\leq t\leq \theta_\infty$ be given by $$ X(t)=X_k(t),\quad \theta_{k-1}\leq t< \theta_k, \quad k\geq k_0. $$ Using \eqref{e1.15} one gets $X(t\wedge\theta_k)=X_k(t\wedge\theta_k)$. It follows from \eqref{e1.14} that \begin{align*} X(t\wedge\theta_k) &=x_a+\int_a^{t\wedge\theta_k} f_k(\tau, X(\tau_-)) \nabla\tau+\int_a^{t\wedge\theta_k}g_k(\tau, X(\tau_-)) \nabla M_\tau \\ &=x_a+\int_a^{t\wedge\theta_k}f(\tau, X(\tau_-))\nabla\tau +\int_a^{t\wedge\theta_k}g(\tau, X(\tau_-))\nabla M_\tau, \end{align*} for any $t\in [a, T]$ and $k\geq 1$. We show that $\lim_{k\to\infty}\theta_k=T$ a.s. Indeed, by \eqref{e1.9} it yields \begin{align*} \mathbb{E}[V(\theta_k\wedge t, X(\theta_k\wedge t))] &=V(a,x_a) +\mathbb{E} \int_a^{t\wedge\theta_k}\Big({V^{\nabla_\tau}}(\tau,X({\tau_-})) +\mathcal{A}V(\tau,X({\tau_-}))\Big)\nabla\tau \\ &\leq V(a,x_a)+ \int_a^{t}( c\mathbb{E} V(\theta_k\wedge \tau_-,X({\theta_k\wedge\tau_-}))+b)\nabla\tau. \end{align*} Using Lemma \ref{lem1.1} with the function $u(t)=\mathbb{E}[V(\theta_k\wedge t, X(\theta_k\wedge t))]+\frac{b}c$ gets \begin{equation*} \mathbb{E} V(\theta_n\wedge t, X(\theta_n\wedge t)) \leq \big(V(a,x_a)+\frac{b}c\big)e_{c}(t,a). \end{equation*} On the other hand, on the set $\{\theta_\inftyC^{-1}_i >\sum_{j\in \mathbb{Z}\setminus\{0\}}|j|^{-5}>1$ and $\mathbb{E} [\xi_2\mid \xi_1]=0$. Therefore, the sequence $M_1=\xi_1$ and $M_2=\xi_1+\xi_2$ is a martingale. Further, \begin{align*} \mathbb{E} [\xi_2^2\mid \xi_1=i] &=\sum_{j\in \mathbb{Z}\setminus\{0\}} j^2\mathbb{P}[\xi_2= j\mid \xi_1= i]\\ &=C_i \sum_{j\in \mathbb{Z}\setminus\{0\}}\frac{j^2}{|j|^{4+\frac{1}{|i|}}} \leq C_i \sum_{j\in \mathbb{Z}\setminus\{0\}} \frac1{|j|^{2}}. \end{align*} Thus $\langle M\rangle_t$ is bounded. On the other hand, \begin{align*} \mathbb{E} |\xi_2|^3 &=\sum_{i,j\in \mathbb{Z}\setminus\{0\}}|j|^3\mathbb{P}[\xi_2= j\mid \xi_1= i] \mathbb{P}\{\xi_1=i\} \\ &=k \sum_{i,j\in \mathbb{Z}\setminus\{0\}}C_i\frac 1{|j|^{(1+\frac{1}{|i|})}|i|^5}\\ &\leq 4k\sum_{i\in \mathbb{Z}\setminus\{0\}}\frac 1{i^4}<\infty, \end{align*} which implies $$ \mathbb{E} |M_1|^3< \infty,\; \mathbb{E} |M_2|^3 \leq 4 (\mathbb{E} \xi_1^3+\mathbb{E} \xi_2^3)<\infty. $$ Consider the dynamic equation on the time scale $\mathbb{T}=\{1,2\}$ \begin{gather*} d^\nabla X_{t}=-X_{t_-}d^{\nabla}t+X_{t_-}d^\nabla M_t \\ X_1=\xi_1. \end{gather*} This equation has a unique solution $X_1=\xi_1$ and $X_2=\xi_1\xi_2$. However, \begin{align*} \mathbb{E} |X_2|^3 &=\mathbb{E} |\xi_1\xi_2|^3 =\sum_{i,j\in \mathbb{Z}\setminus\{0\}}|ij|^3 \mathbb{P}[\xi_2= j\mid \xi_1= i]\mathbb{P}\{\xi_1=i\} \\ &\geq \frac {3k}8\sum_{i,j\in \mathbb{Z}\setminus\{0\}} \frac 1{i^2|j|^{1+\frac{1}{|i|}}}\\ &\geq \frac {3k}8 \sum_{i\in \mathbb{Z}\setminus\{0\}}|i|\frac 1{i^2}=\infty. \end{align*} \end{example} In the following we give conditions ensuring the finiteness of $p$-moment of the solution of \eqref{e1.10}. \begin{theorem}\label{thm1.6} Suppose that linear growth condition \eqref{e1.16} and the conditions \eqref{e1.4}, \eqref{e1.5} hold. Further, there are two constants $m_1, m_p$ such that \begin{equation}\label{e1.17} \int_{\mathbb{R}}|u|{\widehat{\Upsilon}}(t, du)\leq m_1,\quad \int_{\mathbb{R}}|u|^p{\Upsilon}(t, du)\leq m_p\quad \forall t\in [a, T] \end{equation} almost surely. Then, the solution $X_{a,x_a}(t)$ of \eqref{e1.10} starting in $x_a$ satisfies the estimate \begin{equation}\label{e1.18} \mathbb{E}\|X_{a,x_a}(t)\|^p\leq (\|x_a\|^p+1)e_{H}(t,a),\quad a\leq t\leq T \end{equation} where $H$ is a constant. \end{theorem} \begin{proof} Since $\int_{\mathbb{R}}|u|^2{\Upsilon}(t, du)=\langle M \rangle_t\leq N:=m_2$, we can suppose that $p\geq 2$. Applying \eqref{e1.8} to the Lyapunov function $V(t, x)=\|x\|^{p}$ we have \begin{align*} \mathcal{A}V(t,x) &=p\|x\|^{p-2}(1-1_{\mathbb{I}}(t))x^Tf(t,x) +(\|x+f(t,x)\nu(t)\|^{p}-\|x\|^{p})\Phi(t) \\ &\quad +\frac{p}{2}\|x\|^{p-2}\|g(t,x)\|^2\widehat{K}^c_t+\frac{p(p-2)}{2} \|x\|^{p-4}|x^Tg(t,x)|^2\widehat{K}^c_t \\ &\quad +\int_{\mathbb{R}} [\|x+f(t,x)\nu(t)+g(t,x)u\|^{p} -\|x+f(t,x)\nu(t)\|^{p}]{\Upsilon}(t, du) \\ &\quad -p\|x\|^{p-2}x^Tg(t,x)\int_{\mathbb{R}}u\widehat{\Upsilon}(t, du). \end{align*} Using Taylor's expansion for the function $\|x+y\|^{p}$ at $y=0$ obtains \begin{align*} &\|x+f(t,x)\nu(t)\|^{p}-\|x\|^{p}\\ &=p\|x\|^{p-2}x^\top f(t,x)\nu(t) + \frac p2\|x+\theta f(t,x)\nu(t)\|^{p-2}\|f(t,x)\|^2\nu(t)^2\\ &\quad +\frac{p(p-2)}2\|x+\theta f(t,x)\nu(t)\|^{p-4}|(x+\theta f(t,x)\nu(t))^\top f(t,x)|^2\nu(t)^2. \end{align*} where $0\leq\theta\leq 1$. It is seen that \begin{align*} \|x+\theta f(t,x)\nu(t)\|^{p-2}\|f(t,x)\|^2 &\leq (\|x\|+\|f(t,x)\nu(t)\|)^{p-2}\|f(t,x)\|^2 \\ &\leq (\sqrt{1+\|x\|^{2}}+\sqrt {G(1+\|x\|^2)}\nu_*)^{p-2}G(1+\|x\|^2)\\ &=G(1+\sqrt G \nu_*)^{p-2}(1+\|x\|^2)^{p/2}, \end{align*} and \begin{align*} &\|x+\theta f(t,x)\nu(t)\|^{p-4}|(x+\theta f(t,x)\nu(t))^\top f(t,x)|^2 \\ &\leq \|x+\theta f(t,x)\nu(t)\|^{p-2}\| f(t,x)\|^2\\ &\leq G(1+\sqrt G \nu_*)^{p-2}(1+\|x\|^2)^{p/2}. \end{align*} Similarly, the Taylor's expansion of the function $\|x+f(t,x)\nu(t)+y\|^{p}$ at $y=0$ leads us \begin{align*} &\|x+f(t,x)\nu(t)+g(t,x)u\|^{p}-\|x+f(t,x)\nu(t)\|^{p}\\ &=p\|x+f(t,x)\nu(t)\|^{p-2}(x+f(t,x)\nu(t))^\top g(t,x)u\\ &\quad + \frac p2\|x+f(t,x)\nu(t)+\eta g(t,x)u\|^{p-2}\|g(t,x)u\|^2\\ &\quad +\frac{p(p-2)}2\|x+f(t,x)\nu(t)+\eta g(t,x)u\|^{p-4}\\ &\quad\times |(x+f(t,x)\nu(t) +\eta g(t,x)u)^\top g(t,x)u|^2, \end{align*} where $0\leq\eta\leq 1$. By defining $c_p =2^{p-1}$ \text{ if } $p>1$ and $c_p=1$ \text{ if } $p\leq 1$ we have \begin{align*} &\|x+f(t,x)\nu(t)\|^{p-2}(x+f(t,x)\nu(t))^\top g(t,x)u \\ &\leq\|x+f(t,x)\nu(t)\|^{p-1} \|g(t,x)u\|\\ &\leq \sqrt G(1+\sqrt G \nu_*)^{p-1}|u|(1+\|x\|^2)^{p/2}, \end{align*} and \begin{align*} &\|x+f(t,x)\nu(t)+\eta g(t,x)u\|^{p-2}\|g(t,x)u\|^2\\ &\leq c_{p-2}(\|x+f(t,x)\nu(t)\|^{p-2}+(\|g(t,x)\|u)^{p-2})\|g(t,x)u\|^2\\ &\leq c_{p-2}(G(1+\sqrt G \nu_*)^{p-2}u^2+G^{p/2}|u|^p)(1+\|x\|^2)^{p/2}. \end{align*} Further, \begin{align*} &\|x+f(t,x)\nu(t)+\eta g(t,x)u\|^{p-4}|(x+f(t,x)\nu(t)+\eta g(t,x)u)^\top g(t,x)u|^2\\ &\leq \|x+f(t,x)\nu(t)+\eta g(t,x)u\|^{p-2}\|g(t,x)u\|^2\\ &\leq c_{p-2}(G(1+\sqrt G \nu_*)^{p-2}u^2+G^{p/2}|u|^p)(1+\|x\|^2)^{p/2}. \end{align*} Therefore, by using \eqref{bs}, \eqref{e1.17} we obtain \begin{align*} &\mathcal{A}V(t,x)\\ &=p\|x\|^{p-2}(1-1_{\mathbb{I}}(t))x^Tf(t,x)\\ &\quad +(p\|x\|^{p-2}x^\top f(t,x)\nu(t) + \frac p2\|x+\theta f(t,x)\nu(t)\|^{p-2}\|f(t,x)\|^2\nu(t)^2\\ &\quad +\frac{p(p-2)}2\|x+\theta f(t,x)\nu(t)\|^{p-4}|(x+\theta f(t,x)\nu(t))^\top f(t,x)|^2\nu(t)^2)\Phi(t) \\ &\quad +\frac{p}{2}\|x\|^{p-2}\|g(t,x)\|^2\widehat{K}^c_t+\frac{p(p-2)}{2} \|x\|^{p-4}|x^Tg(t,x)|^2\widehat{K}^c_t \\ &\quad +p \int_{\mathbb{R}}\|x+f(t,x)\nu(t)\|^{p-2}(x+f(t,x)\nu(t))^\top g(t,x)u{\widetilde{\Upsilon}}(t, du)\\ &\quad +p \int_{\mathbb{R}}\|x+f(t,x)\nu(t)\|^{p-2}(x+f(t,x)\nu(t))^\top g(t,x)u{\widehat{\Upsilon}}(t, du)\\ &\quad +\frac p2 \int_{\mathbb{R}}\|x+f(t,x)\nu(t) +\eta g(t,x)u\|^{p-2}\|g(t,x)u\|^2{\Upsilon}(t, du)\\ &\quad +\frac{p(p-2)}2\int_{\mathbb{R}}\|x+f(t,x)\nu(t) +\eta g(t,x)u\|^{p-4}|(x+f(t,x)\nu(t)\\ &\quad +\eta g(t,x)u)^\top g(t,x)u|^2{\Upsilon}(t, du) -p\|x\|^{p-2}x^Tg(t,x)\int_{\mathbb{R}}u\widehat{\Upsilon}(t, du) \\ &\leq\Big\{p\sqrt G (1+m_1) +\frac{p(p-1)}{2} G(N +(1+\sqrt G\nu_*)^{p-2}(\nu_*+c_{p-2}N)) \\ &\quad +p\sqrt G(1+\sqrt G\nu_*)^{p-1}m_1 +c_{p-2}\frac{p(p-1)}{2}G^{p/2}m_p\Big\}(1+\|x\|^2)^{p/2}\\ &\leq HV(x) \end{align*} where $H$ is defined \begin{equation}\label{e2.11bs} \begin{aligned} H&= c_{p/2}\Big\{p\sqrt G (1+m_1) +\frac{p(p-1)}{2}G\Big(N +(1+\sqrt G\nu_*)^{p-2}(\nu_*+c_{p-2}N)\Big) \\ &\quad +p\sqrt G(1+\sqrt G\nu_*)^{p-1}m_1 +c_{p-2}\frac{p(p-1)}{2}G^{p/2}m_p\Big\}. \end{aligned} \end{equation} By Theorem \ref{thm1.2}, we obtain $$ \mathbb{E}\|X_{a,x_a}(t)\|^p\leq(\|x_a\|^p+1)e_{H}(t,a), \quad a\leq t\leq T. $$ The proof is complete. \end{proof} \section{Exponential $p$-stability}\label{S2} By \eqref{e1.1}, the $\Delta$-exponential function $e_p$ is also a solution of a $\nabla$-dynamic equations. Therefore, in the following, instead of using $\widehat e_p$, we use $e_p$ to define the exponential stability although we are working with stochastic $\nabla$-dynamic equations. Let the process $K_t$ be bounded on $\mathbb{T}_a$, i.e., the constant $N$ in \eqref{e1.4} does not depend on $T>a$. Suppose that for any $s \geq a; x_s\in \mathbb{R}^d$, the solution $X_{s, x_s}(t)$ with initial condition $X_{s, x_s}(s)=x_s$ of \eqref{e1.10} exists uniquely and it is defined on $\mathbb{T}_s$. Further, \begin{equation}\label{e2.1} f(t, 0)\equiv 0; \quad g(t, 0)\equiv 0. \end{equation} This assumption implies that \eqref{e1.10} has the trivial solution $X_{s, 0}(t)\equiv 0$. \begin{definition} \rm The trivial solution of \eqref{e1.10} is said to be exponentially $p$-stable if there is a positive constant $\alpha$ such that for any $s>a$ there exists $\Gamma=\Gamma(s)>1$, such that \begin{equation}\label{e2.2} \mathbb{E}\|X_{s,x_s}(t)\|^p\leq \Gamma\|x_s\|^pe_{\ominus\alpha}(t, s)\quad \text{on } t\geq s, \end{equation} holds for all $x_s\in\mathbb{R}^d$. \end{definition} If one can choose $\Gamma$ independent of $s$, the trivial solution of \eqref{e1.10} is said to be uniformly exponentially $p$-stable. \begin{remark} \rm Since $\ominus \alpha(t) \leq -\frac{\alpha}{1+\alpha \nu_*}$ for all $t\in \mathbb{T}$, $00$ then $\lim_{t\to\infty}e_{\ominus\alpha}(t,s)=0$. The advantage of using $e_{\ominus\alpha}(t,s)$ is that the requirement $-\alpha \in \mathcal R^+$ is not necessary. \end{remark} \begin{theorem}\label{thm2.2} Suppose that there exist a function $V(t,x)\in C^{1,2}(\mathbb{T}_a\times\mathbb{R}^d; \mathbb{R}_+)$, positive constants $\alpha_1,\alpha_2, \alpha_3$ such that \begin{gather}\label{e2.3} \alpha_1\|x\|^p\leq V(t, x)\leq \alpha_2\|x\|^p, \\ \label{e2.4} { V^{\nabla_t}(t,x)}+ \mathcal{A}V(t, x)\leq -\alpha_3V(t_-, x)\quad \forall (t,x)\in\mathbb{T}_a\times\mathbb{R}^d, \end{gather} where the differential operator $\mathcal{A}$is defined with respect to \eqref{e1.10}. Then, the trivial solution $x\equiv 0$ of \eqref{e1.10} is uniformly exponentially $p$-stable. \end{theorem} \begin{proof} Let $\alpha$ be a positive number satisfying $\frac \alpha{1+\alpha\nu(t)}<\alpha_3$ for all $t\in \mathbb{T}$ and let $s\geq a, x_s\in \mathbb{R}^d$. To simplify notations, we write $X(t)$ for $X_{s,x_s}(t)$. For each $n> \|x_s\|$, define the stopping time $$ \theta_n=\inf\{t\geq s: \quad \|X(t)\|\geq n\}. $$ Obviously, $\theta_n\to\infty$ as $n\to\infty$ almost surely. By \eqref{e2.9}, calculating expectations we obtain \begin{align*} &\mathbb{E}[e_{\alpha}(t\wedge\theta_n, s)V(t\wedge\theta_n, X(t\wedge\theta_n))]\\ &=V(s,x_s) +\mathbb{E}\int_s^{t\wedge\theta_n} e_{\alpha}(\theta_n\wedge \tau_-, s)\Big[\alpha V(\tau_-, X(\tau_-))\\ &\quad +(1+\alpha\nu(\tau))(V^{\nabla_\tau}(\tau,X(\tau_-)) + \mathcal{A}V(\tau,X(\tau_-)))\Big]\nabla\tau. \end{align*} Using \eqref{e2.4} and the inequality $\frac \alpha{1+\alpha\nu(t)}<\alpha_3$ obtains $$ \alpha V(\tau_-, X(\tau_-)) +(1+\alpha\nu(\tau))\big(V^{\nabla_\tau}(\tau,X(\tau_-)) + \mathcal{A}V(\tau,X(\tau_-))\big)\leq0. $$ Therefore, \begin{align*} \alpha_1e_{\alpha}(t\wedge\theta_n, s)\mathbb{E} \|X(t\wedge\theta_n)\|^p &\leq \mathbb{E}[e_{\alpha}(t\wedge\theta_n, s)V(t\wedge\theta_n, X(t\wedge\theta_n))]\\ &\leq V(s,x_s)\leq \alpha_2\|x_s\|^p. \end{align*} Letting $n\to\infty$ yields $$ \alpha_1e_{\alpha}(t, s)\mathbb{E} \|X(t)\|^p\leq \alpha_2\|x_s\|^p. $$ Hence, $$ \mathbb{E} \|X_{s,x_s}(t)\|^p \leq \frac{\alpha_2}{\alpha_1}\|x_s\|^pe_{\ominus \alpha}(t, s). $$ The proof is complete. \end{proof} We now consider the inverse problem by showing that if the trivial solution of \eqref{e1.10} is uniformly exponentially $p$-stable then such a Lyapunov function exits. Firstly, we study the differentiability of solutions with respect to the initial conditions and the continuity with respect to coefficients. \begin{lemma}[Burkholder inequality on time scales]\label{lem2.3} For any $p\geq 2$ there exist positive constants $B_p$ such that if $\{M_t\}_{t\in\mathbb{T}_a}$ is an $\mathcal{F}_t$-martingale with $\mathbb{E}|M_t|^p<\infty$ and $ M_a=0$ then \begin{equation*} \mathbb{E}\sup_{a\leq s\leq t}|M_s|^p\leq B_p \Big(\mathbb{E}\langle M\rangle_t^{p/2} +\mathbb{E}\sum_{a\leq s\leq t}|\nabla^*M_s|^p\Big), \end{equation*} where $\nabla^*M_s=M_s-M_{s_-}$. \end{lemma} \begin{proof} By Doob's inequality, we have $$ \mathbb{E}\sup_{a\leq s\leq t}|M_s|^p\leq\Big(\frac{p}{p-1}\Big)^p\mathbb{E}|M_t|^p. $$ Otherwise, we see that the martingale $\widehat M_t$ can be extended to a regular martingale on $[a; \infty)_{\mathbb{R}}$. Therefore, by using proof of \cite[Lemma 5]{Tar} we obtain $$ \mathbb{E}|\widehat{M}_t|^p \leq \widehat{B}_p \Big(\mathbb{E}\langle \widehat{M}\rangle_t^{p/2} +\mathbb{E}\sum_{a\leq s\leq t}|\nabla^*\widehat{M}_s|^p\Big), $$ for a constant $\widehat{B}_p$. Further, the martingale $\widetilde{M}_t$ is a sum of random variables. Then, applying \cite[Theorem 13.2.15, pp.416]{Ath} yields \begin{equation*} \mathbb{E}|\widetilde{M}_t|^p \leq \widetilde{B}_p\Big(\mathbb{E}\langle \widetilde{M}\rangle_t^{p/2} +\mathbb{E}\sum_{a\leq s\leq t}|\nabla^*\widetilde{M}_s|^p\Big). \end{equation*} Consequently, \begin{align*} \mathbb{E}\sup_{a\leq s\leq t}|M_s|^p &\leq 2^{p-1}\Big(\frac{p}{p-1}\Big)^p \Big(\mathbb{E}|\widehat{M}_t|^p+\mathbb{E}|\widetilde{M}_t|^p\Big)\\ &\leq 2^{p-1}\Big(\frac{p}{p-1}\Big)^p\Big[\widehat{B}_p \Big(\mathbb{E}\langle \widehat{M}\rangle_t^{p/2} +\mathbb{E}\sum_{a\leq s\leq t}|\nabla^*\widehat{M}_s|^p\Big) +\widetilde{B}_p\Big(\mathbb{E}\langle \widetilde{M}\rangle_t^{p/2}\\ &\quad +\mathbb{E}\sum_{a\leq s\leq t}|\nabla^*\widetilde{M}_s|^p\Big)\Big] \\ &\leq B_p\Big(\mathbb{E}(\langle \widehat{M}\rangle_t +\langle \widetilde{M}\rangle_t)^{p/2} +\mathbb{E}\sum_{a\leq s\leq t}(|\nabla^*\widehat{M}_s|^p +|\nabla^*\widetilde{M}_s|^p)\Big) \\ &=B_p\Big(\mathbb{E}\langle M\rangle_t^{p/2} +\mathbb{E}\sum_{a\leq s\leq t}|\nabla^*{M}_{s}|^p\Big) \end{align*} where $B_p=2^{p}(\frac{p}{p-1})^p \max\big\{\widehat{B}_p, \widetilde{B}_p\big\}$. The proof is complete. \end{proof} \begin{theorem}\label{thm2.4} Let $p\geq 2, M\in\mathcal{M}_2$ such that the conditions \eqref{e1.4}, \eqref{e1.5} and \eqref{e1.17} hold and let $ g\in\mathcal{L}_2((a,T];M)$ with $$ \int_a^t\mathbb{E}|g(\tau)|^p\nabla\tau<\infty \;\; \forall t\in \mathbb{T}_a. $$ Then \begin{equation*} \mathbb{E}\sup_{a\leq t\leq T}\Big|\int_a^t g(\tau)\nabla M_\tau\Big|^p \leq C_p \int_a^T\mathbb{E}|g(\tau)|^p\nabla\tau, \end{equation*} where $C_p=B_p\{(T-a)^{\frac{p}{2}-1}N^{p/2}+ m_p\}$. \end{theorem} \begin{proof} Set $$ x_t=\int_a^tg(\tau)\nabla M_\tau,\quad t\in[a, T]. $$ The process $x_t$ is a square martingale with the characteristic $$ \langle x\rangle_t=\int_a^t |g(\tau)|^2\nabla \langle M\rangle_\tau. $$ Since $\langle M\rangle_t$ is continuous, so is $\langle x\rangle_t$. Applying Lemma \ref{lem2.3} to the martingale $(x_t)$ obtains \begin{align*} &\mathbb{E}\sup_{a\leq r\leq t}|x_r|^p\\ &\leq B_p\Big\{\mathbb{E}\langle x\rangle_t^{p/2} +\mathbb{E}\sum_{a\leq s\leq t}|\nabla^*x_{s}|^p\Big\} \\ &= B_p\Big\{\mathbb{E}\langle x\rangle_t^{p/2} +\mathbb{E}\int_a^t\int_{\mathbb{R}} |g(\tau)u|^p\delta(\nabla \tau,du)\Big\} \\ &= B_p\Big\{\mathbb{E}\Big[\int_a^t|g(\tau)|^2\nabla\langle M\rangle_\tau\Big]^{p/2} +\mathbb{E}\int_a^t\int_{\mathbb{R}} |g(\tau)u|^p\pi(\nabla\tau,du)\Big\} \\ &\leq B_p\Big\{(t-a)^{\frac{p}{2}-1}N^{p/2}\int_a^t\mathbb{E}|g(\tau)|^p\nabla\tau +\mathbb{E}\int_a^t|g(\tau)|^p\int_{\mathbb{R}}|u|^p\Upsilon(\tau, du)\nabla\tau\Big\} \\ &\leq B_p\big\{(t-a)^{\frac{p}{2}-1}N^{p/2}+ m_p\big\} \int_a^t\mathbb{E}|g(\tau)|^p\nabla\tau. \end{align*} By putting $C_p=B_p\big[(T-a)^{\frac{p}{2}-1}N^{p/2}+ m_p\big]$ we complete the proof. \end{proof} \begin{lemma}\label{lem2.5'} Let $T,s\in\mathbb{T}_a; T>s$ and $p\geq2$ fixed. Suppose that the condition \eqref{e1.17} holds and process $\zeta(t)$ is the solution of the stochastic equation \begin{equation}\label{e3.5'} \zeta(t)=\varphi(t)+\int_s^t\psi(\tau)\zeta(\tau_-)\nabla\tau+ \int_s^t\chi(\tau)\zeta(\tau_-)\nabla M_\tau, \;\; \forall t\in[s, T]. \end{equation} We assume that the functions $ \varphi(t), \psi(t)$ and $\chi(t)$ are $\mathcal{F}_t$-adapted and that there exists a constant $K>0$ such that with probability $1$, $\|\psi(t)\| \leq K$ and $\| \chi (t) \| \leq K$. Then \begin{equation}\label{2.6b2} \mathbb{E}\sup_{s\leq t\leq T} \|\zeta (t)\|^p\leq 3^{p-1}\mathbb{E} \sup_{s\leq t\leq T} \|\varphi(t)\|^p e_{H_1}(T,s), \end{equation} where $H_1=3^{p-1}K^p((T-s)^{p-1} +C_p)$. \end{lemma} \begin{proof} For any $n>0$ denote $\theta_n=\inf\{t>s: \|\zeta(t)\|>n\}$. From \eqref{e3.5'} we have \begin{align*} &\mathbb{E}\sup_{s\leq r\leq t}\|\zeta(r\wedge \theta_n)\|^p\\ &\leq 3^{p-1}\Big(\mathbb{E}\sup_{s\leq r\leq T}\|\varphi(r)\|^p +\mathbb{E}\sup_{s\leq r\leq t}\Big\|\int_s^{r\wedge \theta_n} \psi(\tau)\zeta(\tau_-)\nabla\tau\Big\|^p\\ &\quad + \mathbb{E}\sup_{s\leq r\leq t} \Big\|\int_s^{r\wedge \theta_n}\chi(\tau)\zeta(\tau_-)\nabla M_\tau\Big\|^p\Big)\\ &\leq 3^{p-1}\Big(\mathbb{E}\sup_{s\leq r\leq T}\|\varphi(r)\|^p +K^p(T-a)^{p-1}\int_s^{t\wedge \theta_n}\mathbb{E}\| \zeta(\tau_-)\|^p\nabla\tau\\ &\quad + C_pK^p\int_s^{t\wedge \theta_n}\mathbb{E}\|\zeta(\tau_-)\|^p\nabla \tau\Big) \quad \text{(by Theorem \ref{thm2.4})} \\ &=3^{p-1}\Big(\mathbb{E}\sup_{s\leq r\leq T}\|\varphi(r)\|^p+K^p\Big((T-a)^{p-1} +C_p\Big)\int_s^{t\wedge \theta_n}\mathbb{E}\|\zeta(\tau_-)\|^p\nabla \tau\Big)\\ &=3^{p-1}\mathbb{E}\sup_{s\leq r\leq T}\|\varphi(r)\|^p+H_1\int_s^{t} \sup_{s\leq r\leq \tau_-}\mathbb{E}\|\zeta(r\wedge \theta_n)\|^p\nabla \tau, \end{align*} where $H_1=3^{p-1}K^p((T-s)^{p-1} +C_p)$. Using Lemma \ref{lem1.1} one gets \begin{equation*} \mathbb{E}\sup_{s\leq t\leq T} \|\zeta (t\wedge \theta_n)\|^p\leq 3^{p-1}\mathbb{E}\sup_{s\leq t\leq T} \|\varphi(t)\|^p e_{H_1}(T,s). \end{equation*} Letting $n\to\infty$ yields \eqref{2.6b2}. The proof is complete. \end{proof} \begin{lemma}\label{lem2.5} Suppose that the coefficients of \eqref{e1.10} are continuous in $s, x$ and they have continuous bounded first and second partial derivatives and condition \eqref{e1.17} holds for $p\geq 4$. Then, the solution $X_{s,x}( t)$, $s\leq t\leq T$, with initial condition $X_{s,x}( s)=x$ of \eqref{e1.10} is twice differentiable with respect to $x$. Further, the derivatives \[ \frac{\partial}{\partial x_i}(X_{s, x}(t)),\quad \frac{\partial^2}{\partial x_i\partial x_j}(X_{s, x}(t)) \] are continuous in $x$ in mean square. \end{lemma} \begin{proof} Suppose that the derivatives $f_x'(t,x), g_x'(t,x),f_{xx}''(t,x), g_{xx}''(t,x) $ are bounded by a constant $\lambda$. To simplify notations we put $Y_{s,\Delta x}(t)=X_{s, x+\Delta x}(t) - X_{s, x}(t)$. Using Lagrange theorem we see that for any $i=1,2,\dots,d$, there exists $ \theta_i, \xi_i\in[0; 1]$ such that \begin{equation}\label{e2.6b1} \begin{aligned} &f_i(t,X_{s, x}(t_-)+Y_{s, \Delta x}(t_-))-f_i(t,X_{s, x}(t_-))\\ &=\sum_{j=1}^d\frac {\partial f_i}{\partial x_j}(t,X_{s, x}(t_-) +\theta_iY_{s, \Delta x}(t_-))Y_{i,s, \Delta x}(t_-), \\ &g_i(t,X_{s, x}(t_-)+Y_{s, \Delta x}(t_-))-g_i(t,X_{s, x}(t_-)) \\ &=\sum_{j=1}^d\frac {\partial g_i}{\partial x_j}(t,X_{s, x}(t_-) +\xi_iY_{s, \Delta x}(t_-))Y_{i,s, \Delta x}(t_-). \end{aligned} \end{equation} Let $A_{s,\Delta x}(t)$ be the matrix with entries $a^{ij}_{s,\Delta x}(t)=\frac {\partial f_i}{\partial x_j}(t,X_{s, x}(t_-) +\theta_iY_{s, \Delta x}(t_-))$, and let $B_{s,\Delta x}(t)$ be the matrix with entries $b^{ij}_{s,\Delta x}(t) =\frac {\partial g_i}{\partial x_j}(t, X_{s, x}(t_-)+\xi_iY_{s, \Delta x}(t_-))$. Then \eqref{e2.6b1} can be rewritten \begin{gather*} f(t, X_{s, x}(t_-)+Y_{s, \Delta x}(t_-))-f(t, X_{s, x}(t_-)) =A_{s, \Delta x}(t)Y_{s, \Delta x}(t_-),\\ g(t, X_{s, x}(t_-)+Y_{s, \Delta x}(t_-))-g(t, X_{s, x}(t_-)) =B_{s, \Delta x}(t)Y_{s, \Delta x}(t_-). \end{gather*} Hence, \begin{align*} Y_{s,\Delta x}(t)= \Delta x +\int_s^tA_{s, \Delta x}(\tau)Y_{s, \Delta x}(\tau_-)\nabla \tau +\int_s^tB_{s, \Delta x}(\tau)Y_{s, \Delta x}(\tau_-)\nabla M_\tau. \end{align*} Since $A_{s, \Delta x}(t)$ and $B_{s, \Delta x}(t)$ are bounded by a constant $\lambda$, by using Lemma \ref{lem2.5'} one has \begin{equation}\label{e2.6b} \mathbb{E}\sup_{s\leq t\leq T}\|Y_{s,\Delta x}(t)\|^{2} \leq 3\|\Delta x\|^{2} e_{H_2}(T, s), \end{equation} where $H_2=3\lambda^2(T-s +C_2)$. As a consequence, $ \mathbb{E}\sup_{s\leq t\leq T}\|Y_{s,\Delta x}(t)\|^{2}\to 0$ as $\|\Delta x\|\to 0$ in probability. Let $\zeta_{s,x}(t)$ be the solution of the variation dynamic equation \[ \zeta_{s,x}(t) = I+\int_s^tf'_x(\tau,X_{s, x}(\tau_-))\zeta_{s,x}(\tau_-)\nabla\tau +\int_s^tg'_x(\tau, X_{s, x}(\tau_-)\zeta_{s,x}(\tau_-)\nabla M_\tau, \] for all $ s\leq t\leq T$. Since $f'_x$ and $g'_x$ are bounded by constant $\lambda$, \begin{equation}\label{e2.7b1} \mathbb{E}\sup_{s\leq t\leq T}\|\zeta_{s,x}(t)\|^4 \leq 27e_{H_3}(T,s), \end{equation} where $H_3=27\lambda^4((T-s)^{3}+C_4)$. Define $$ \zeta_{\Delta x}(t)=Y_{s,\Delta x}(t)-\zeta_{s,x}(t){\Delta x}\;\;\;\forall \, s\leq t\leq T. $$ The process $\zeta_{\Delta x}(t)$ satisfies Equation \[ \zeta_{\Delta x}(t) =\phi_{\Delta x}(t)+\int_s^tA_{s,\Delta x}(\tau)\zeta_{\Delta x}(\tau_-) \nabla \tau+ \int_s^tB_{s,\Delta x}(\tau)\zeta_{\Delta x}(\tau_-)\nabla M_\tau, \] where, \begin{align*} \phi_{\Delta x}(t) &=\int_s^t[(A_{s, \Delta x}(\tau)-f'_x(\tau,X_{s, x}(\tau_-)))\zeta_{s,x} (\tau_-)\Delta x]\nabla \tau \\ &\quad +\int_s^t[(B_{s, \Delta x}(\tau)-g'_x(\tau,X_{s, x}(\tau_-))) \zeta_{s,x}(\tau_-)\Delta x]\nabla M_\tau. \end{align*} Applying Lemma \ref{lem2.5'} again one gets \begin{equation}\label{e2.6} \mathbb{E}\sup_{s\leq t\leq T}\|\zeta_{\Delta x}(t)\|^2 \leq 3\mathbb{E}\sup_{s\leq t\leq T}\|\phi_{\Delta x}(t)\|^2e_{H_2}(T, s). \end{equation} Since $f_x'(t,x), g_x'(t,x)$ are continuous and $ \mathbb{E}\sup_{s\leq t\leq T}\|Y_{s,\Delta x}(t)\|^{2}\to 0$ as $\|\Delta x\|\to 0$ in probability, $$ \lim_{\Delta x\to 0}(\|A_{s, \Delta x}(t)-f'_x(t ,X_{s, x}(t_-))\| +\|B_{s, \Delta x}(t)-g'_x(t ,X_{s, x}(t_-))\|)=0 $$ in probability. Hence, by the boundedness of $A,B,f', g'$, we obtain \begin{equation}\label{e3.10b} \begin{aligned} &\mathbb{E}\big[\sup_{s\leq t\leq T}\frac{\|\phi_{\Delta x}(t)\|^2}{\|\Delta x\|^2} \big]\\ &\leq 2(T-s)\int_s^T \mathbb{E}\|A_{s, \Delta x}(\tau) -f'_x(\tau,X_{s, x}(\tau_-))\zeta_{s,x}(\tau_-)\|^2\nabla \tau \\ &\quad +8\int_s^T\mathbb{E}\|B_{s,\Delta x}(\tau)-g'_x(\tau,X_{s, x}(\tau_-))\zeta_{s,x}(\tau_-)\|^2\nabla \langle M\rangle_\tau \to 0 \end{aligned} \end{equation} as $\|\Delta x\|\to 0$. Thus, \eqref{e2.6} and \eqref{e3.10b} imply $$ \mathbb{E}\sup_{t\leq s\leq T}\frac{\|\zeta_{\Delta x}(s)\|}{\|\Delta x\|}= 0 \quad \text{as } \Delta x\to 0. $$ This means $$ \zeta_{s,x}(t)=\frac{\partial}{\partial x}X_{s, x}(t)\quad \forall s\leq t\leq T. $$ The mean square continuity of $\zeta_{s,x}(t)$ with respect to $x$ again follows from the continuity of $f'_x(t, X_{s, x}(t))\quad \text{and}\quad g'_x(t, X_{s, x}(t))$. We prove the existence of $\frac{\partial^2 X_{s,x}(t) }{\partial x^2}$. To simplify notations, if $F$ is a bilinear mapping, we write $F h^2$ for $F(h,h)$. Let bilinear mapping $\eta_{s,x}(t)$ be the solution of the second variation dynamic equation \begin{align*} \eta_{s,x}(t) &=\int^t_sf^{''}_{xx}(\tau, X_{s,x}(\tau_-))\zeta_{s,x}^2(\tau_-)\nabla\tau +\int^t_s f'_x(\tau, X_{s,x}(\tau_-))\eta_{s,x}(\tau_-)\nabla \tau\\ &\quad + \int^t_sg^{''}_{xx}(\tau, X_{s,x}(\tau_-))\zeta_{s,x}^2(\tau_-)\nabla M_\tau + \int^t_s g'_x(\tau, X_{s,x}(\tau_-))\eta_{s,x}(\tau_-)\nabla M_\tau, \end{align*} for all $ s\leq t\leq T$. Using Lemma \ref{lem2.5'} and \eqref{e2.7b1} we see that \begin{equation}\label{e2.12b} \mathbb{E} \sup_{s\le t\leq T}\|\eta_{s,x}(t)\|^2\leq \infty. \end{equation} Define $$ \eta_{\Delta x}(t)=\zeta_{s,x+\Delta x}(t)\Delta x-\zeta_{s,x}(t) \Delta x-\eta_{s,x}(t)(\Delta x)^2,\quad s\leq t\leq T. $$ The process $\eta_{\Delta x}(t)$ satisfies the equation \begin{equation}\label{e2.9} \begin{aligned} \eta_{\Delta x}(t) &=\psi_{\Delta x}(t)+ \int_s^tf'_x(\tau,X_{s, x+\Delta x}(\tau_-)) \eta_{\Delta x}(\tau_-)\nabla \tau \\ &\quad +\int_s^tg'_x(\tau, X_{s, x+\Delta x}(\tau_-)) \eta_{\Delta x}(\tau_-)\nabla M_\tau, \end{aligned} \end{equation} where, \begin{align*} \psi_{\Delta x}(t) &=\int_s^t\Big[\Big(f'_x(\tau,X_{s, x+\Delta x}(\tau_-))-f'_x(\tau,X_{s, x}(\tau_-))\\ &\quad -f''_{xx}(\tau,X_{s, x}(\tau_-)\Big) \zeta_{s,x}(\tau_-)\Delta x\big)\zeta_{s,x}(\tau_-)\Delta x\\ &\quad +(f'_x(\tau,X_{s, x+\Delta x}(\tau_-)) -f'_x(\tau,X_{s, x}(\tau_-)))\eta_{s,x}(\tau_-)(\Delta x)^2\Big]\nabla \tau \\ &\quad +\int_s^t\Big[\big(g'_x(\tau,X_{s, x+\Delta x}(\tau_-)) -g'_x(\tau,X_{s, x}(\tau_-))\\ &\quad -g''_{xx}(\tau,X_{s, x}(\tau_-))\zeta_{s,x}(\tau_-)\Delta x\big) \zeta_{s,x}(\tau_-)\Delta x \\ &\quad +\big(g'_x(\tau,X_{s, x+\Delta x}(\tau_-)) -g'_x(\tau,X_{s, x}(\tau_-))\big)\eta_{s,x}(\tau_-)(\Delta x)^2\Big]\nabla M_\tau. \end{align*} Using Lemma \ref{lem2.5'} one obtains \begin{equation}\label{e2.9b} \mathbb{E}\|\eta_{\Delta x}(t)\|^2 \leq \mathbb{E}\sup_{s\leq t\leq T}\|\psi_{\Delta x}(t)\|^2 e_{H_2}(T,s), \end{equation} where $H_2=3\lambda^2(T-s+4N)$. It is easy to see that \begin{align*} &\mathbb{E}\sup_{s\leq t\leq T}\big\|\int_s^t\Big[\big(f'_x(\tau,X_{s, x+\Delta x}(\tau_-))-f'_x(\tau,X_{s, x}(\tau_-)) -f''_{xx}(\tau,X_{s, x}(\tau_-)) \\ &\times \zeta_{s,x}(\tau_-)\Delta x\big) \zeta_{s,x}(\tau_-)\Delta x\Big]\nabla\tau\big\|^2 \\ &\leq 2(T-s)\mathbb{E}\int_s^T\big\|\big(f'_x(\tau,X_{s, x+\Delta x}(\tau_-))\\ &-f'_x(\tau,X_{s, x}(\tau_-))-f''_{xx}(\tau,X_{s, x}(\tau_-))Y_{s,\Delta x}(\tau_-) \big)\zeta_{s,x}(\tau_-)\Delta x\big\|^2\nabla \tau \\ &\quad +2(T-s)\mathbb{E}\int_s^T\big\|f''_{xx}(\tau,X_{s, x}(\tau_-))(Y_{s,\Delta x} (\tau_-)-\zeta_{s,x}(\tau_-)\Delta x)\\ &\quad \times\zeta_{s,x}(\tau_-)\Delta x\|^2\nabla \tau=o(\|\Delta x\|^4); \end{align*} \begin{align*} &\int_s^T\mathbb{E}\Big\|\big(f'_x(\tau,X_{s, x+\Delta x}(\tau_-)) -f'_x(\tau,X_{s, x}(\tau_-))\big)\eta_{s,x}(\tau_-)(\Delta x)^2\Big\|^2\nabla \tau\\ &=o(\|\Delta x\|^4); \end{align*} \begin{align*} &\mathbb{E}\sup_{s\leq t\leq T}\Big\|\int_s^t \Big[\big(g'_x(\tau,X_{s, x+\Delta x}(\tau_-))-g'_x(\tau,X_{s, x}(\tau_-)) -g''_{xx}(\tau,X_{s, x}(\tau_-))\\ &\times\zeta_{s,x}(\tau_-)\Delta x\big) \zeta_{s,x}(\tau_-)\Delta x\Big]\nabla M_\tau\Big\|^2\\ &\leq 4N\mathbb{E}\int_s^T\big\|\big(g'_x(\tau,X_{s, x+\Delta x}(\tau_-)) \\ &\quad -g'_x(\tau,X_{s, x}(\tau_-))-g''_{xx}(\tau,X_{s, x}(\tau_-)) Y_{s,\Delta x}(\tau_-)\big)\zeta_{s,x}(\tau_-)\Delta x\big\|^2\nabla \tau\\ &\quad +4N\mathbb{E}\int_s^T\big\|g''_{xx}(\tau,X_{s, x}(\tau_-)) (Y_{s,\Delta x}(\tau_-)-\zeta_{s,x}(\tau_-)\Delta x)\zeta_{s,x}(\tau_-) \Delta x\|^2\nabla \tau\\ &=o(\|\Delta x\|^4); \end{align*} \begin{align*} &\mathbb{E}\sup_{s\leq t\leq T}\hskip -.05cm\Big\|\int_s^t \Big[\big(g'_x(\tau,X_{s, x+\Delta x}(\tau_-)) -g'_x(\tau,X_{s, x}(\tau_-))\big)\eta_{s,x}(\tau_-)(\Delta x)^2\Big] \nabla M_\tau\Big\|^2 \\ &\leq 4N\mathbb{E}\int_s^T\big\|(g'_x(\tau,X_{s, x+\Delta x}(\tau_-)) -g'_x(\tau,X_{s, x}(\tau_-)))\eta_{s,x}(\tau_-)(\Delta x)^2\big\|^2\nabla \tau\\ &=o(\|\Delta x\|^4). \end{align*} Combining these results we obtain $\mathbb{E}\sup_{s\leq t\leq T}\|\psi_\Delta(t)\|^2=o(\|\Delta x\|^4)$, which implies that $$ \mathbb{E}\|\eta_{\Delta x}(t)\|^2=o(\|\Delta x\|^4). $$ Thus, $\frac {\|\eta_{\Delta x}(t)\|}{\|\Delta x\|^2}=0$, or $$ \frac {\partial^2}{\partial x^2}X_{s,x}(t)=\eta_{s,x}(t). $$ The proof is complete. \end{proof} \begin{lemma}\label{lem2.6'} Let $p\geq4$ and $2\leq \beta\leq p$. Then, the map $F(\phi):\phi\to \mathbb{E}|\phi|^\beta$ from $L_p(\Omega,\mathcal{F},\mathbb{P})$ to $\mathbb{R}$ is twice differentiable at every $\phi_0\ne 0$ and $$ F'(\phi_0)(\phi)=\beta \mathbb{E}[ |\phi_0|^{\beta-1}\phi];\quad F''(\phi_0)(\phi,\psi)=\beta(\beta-1) \mathbb{E} [|\phi_0|^{\beta-2}\phi\psi].$$ \end{lemma} \begin{proof} We have \begin{align*} &\big|F(\phi_0+\Delta \phi)-F(\phi_0)-\beta \mathbb{E} |\phi_0|^{\beta-1}\Delta\phi \big|\\ &= \big|\mathbb{E} |\phi_0+\Delta\phi|^{\beta} - \mathbb{E} |\phi_0|^{\beta}-\beta \mathbb{E} |\phi_0|^{\beta-1}\Delta\phi\big|\\ &= \beta(\beta-1)\mathbb{E} [|\eta|^{\beta-2}(\Delta\phi)^2]\\ &\leq \beta(\beta-1)[\mathbb{E} |\eta|^{m(\beta-2)}]^{1/m} [\mathbb{E}|\Delta\phi|^p]^{2/p}, \end{align*} where $\eta \in (\phi_0, \phi_0+\Delta \phi)$ if $\phi_0+\Delta \phi>\phi_0$ and $\eta \in (\phi_0+\Delta \phi, \phi_0)$ if $\phi_0+\Delta \phi<\phi_0$. Hence, with $\frac1m+\frac 2p=1$ we have \begin{align*} &\big |F(\phi_0+\Delta \phi)-F(\phi_0)-\beta \mathbb{E} |\phi_0|^{\beta-1} \Delta\phi\big|\\ &\leq \beta(\beta-1)[\mathbb{E} |\eta|^{m(\beta-2)}]^{1/m} [\mathbb{E}|\Delta\phi|^p]^{2/p}\\ &\leq \beta(\beta-1)[\mathbb{E} \max\{|\phi_0|,|\phi_0+\Delta\phi|\}^{m(\beta-2)} ]^{1/m}[\mathbb{E}|\Delta\phi|^p]^{2/p}. \end{align*} The relation $\frac 1m+\frac 2p=1$ implies $m(\beta-2)a$, the function $u(s,x)=\mathbb{E}\|X_{s,x}(t)\|^\beta;\; a]\\ \begin{aligned} u''_{xx}(s,x)h^2 &=\beta\mathbb{E}\Big[(\beta-2)\|X_{s,x}(t)\|^{\beta-4} \langle X_{s,x}(t),\zeta_{s,x}(t)h \rangle^2 \\ &\quad +\|X_{s,x}(t)\|^{\beta-2}\|\zeta_{s,x}(t)h\|^2 +\|X_{s,x}(t)\|^{\beta-2}\langle X_{s,x}(t),\eta_{s,x}(t)h^2\rangle \Big] \end{aligned} \notag. \end{gather} The proof is complete. \end{proof} \begin{theorem}\label{thm2.7} Let $M$ have independent increments and the conditions of Lemma \ref{lem2.5} hold and $2\leq \beta\leq p$. Suppose further that $\mathcal{A}V(t,x)$ is $ld$-continuous in $(t,x)$ for all $V\in C^{1,2}(\mathbb{T}_a\times \mathbb{R}^d; \mathbb{R})$. Then, the function $u(s,x)=\mathbb{E} \|X_{s,x}(t)\|^\beta$, $ a0$, there exist a function $\gamma_T:\mathbb{T}\to\mathbb{T}$ with $\gamma(T, s)\geq s+T$ for all $s\in\mathbb{T}$ such that $\gamma(T, s)$ and $\nabla$-derivatives $\gamma^{\nabla_s}(T, s)$ are bounded. If the trivial solution of \eqref{e1.10} is uniformly exponentially $\beta$-stable, then there exists a function $V(s, x)\in C^{1,2}(\mathbb{T}_a\times\mathbb{R}^d; \mathbb{R}_+)$ satisfying inequalities \eqref{e2.3}, \eqref{e2.4} (with the power $\beta$). \end{theorem} \begin{proof} By Lemma \ref{lem2.6} and Theorem \ref{thm2.7}, the function \begin{equation}\label{e3.16} V(s,x)=\int_s^{\gamma(T, s)}\mathbb{E}\|X_{s,x}(\tau_- )\|^\beta\nabla\tau, \end{equation} is in class $C^{1,2}(\mathbb{T}_a\times\mathbb{R}^d; \mathbb{R}_+)$. From \eqref{e2.2}, $$ V(s_-,x)\leq \int_{s_-}^{\gamma(T, {s_-})}\Gamma\|x\|^\beta e_{\ominus\alpha}(\tau_-, s_-)\nabla\tau\leq\alpha_1\|x\|^\beta, $$ where $\alpha_1=\frac{\Gamma(1+\nu_*\alpha)}{\alpha}$. By assumptions, the trivial solution of \eqref{e1.10} is uniformly exponentially $\beta$-stable and $\gamma^{\nabla_s}(T,s)$ is bounded, we can choose $T>0$ such that \begin{equation}\label{e2.20} \mathbb{E}\|X_{s, x}(\gamma(T, s))\|^\beta<\frac{1}{2}\|x\|^\beta,\quad \mathbb{E}\|X_{s, x}(\gamma(T,s))\|^\beta\gamma^{\nabla_s}(T, s) <\frac{1}{2}\|x\|^\beta. \end{equation} Since $f$ and $g$ have bounded partial derivatives and $f(t,0)=0,\; g(t,0)=0$, $$ \|f(t,x)\|\leq G \|x\|,\quad \|g(t,x)\|\leq G \|x\|, \quad t\geq a,\; x\in \mathbb{R}^d. $$ Therefore, \begin{equation}\label{e2.19} \|\mathcal{A}[\|x\|^\beta](s,x)\|\alpha_2\|x\|^\beta$ with $\alpha_2=\frac{1}{2c_1}$. Thus, the function $V$ satisfies condition \eqref{e2.3}. Using \cite[Theorem 5.80]{Pet} to calculate $\nabla$-differential of $V$ with respect $s$ and applying Theorem \ref{thm2.7} we obtain \begin{equation*} V^{\nabla_s}(s,x)+\mathcal{A}V(s,x) =\mathbb{E}\|X_{s_-, x}(\gamma(T,s_-))\|^\beta\gamma^{\nabla_s}(T, s_-)-\|x\|^\beta. \end{equation*} Using \eqref{e2.20} again we have \begin{equation*} V^{\nabla_s}(s,x)+\mathcal{A}V(s,x) \leq -\frac{1}{2}\|x\|^\beta\leq -\frac{1}{2\alpha_1}V(s_-,x). \end{equation*} Thus, the function $V$ satisfies all conditions \eqref{e2.3}, \eqref{e2.4} with $\alpha_3=\frac{\alpha}{2\Gamma(1+\nu_*\alpha)}$. The proof is complete. \end{proof} \begin{example} \rm Consider the linear stochastic dynamic equation \begin{equation}\label{e2.21} \begin{gathered} d^\nabla X(t)=aX(t_-)d^\nabla t+bX(t_-)d^\nabla M(t) \quad\forall t\in\mathbb{T}_s \\ X(s)=x, \end{gathered} \end{equation} where $a,b$ are two constants, $a$ is regressive and $M$ is a square integrable martingale having independent increment. By direct calculation we have \begin{equation}\label{3.19} \mathbb{E} X^2_{s,x}(t)=x^2+\int_{s}^t{q}(\tau)\mathbb{E} X_{s,x}^2(\tau_-) \nabla\tau, \end{equation} where \begin{align*} q(t)&=2a +b^2 \widehat K_t^c +a^2\nu(t) +2b(1+a\nu(t))\int_{\mathbb{R}}u{\Upsilon}(t, du)\\ &\quad +b^2\int_{\mathbb{R}}u^2{\Upsilon}(t, du) -2b\int_{\mathbb{R}}u{\widehat\Upsilon}(t, du)\\ &=2a +b^2 \widehat K_t^c +a^2\nu(t) +2b\int_{\mathbb{R}}u{\widetilde \Upsilon}(t, du)\\ &\quad +b^2\int_{\mathbb{R}}u^2{\Upsilon}(t, du) +a\nu(s)\int_{\mathbb{R}}u{\Upsilon}(t, du). \end{align*} Since $\int_{\mathbb{R}}u{\widetilde \Upsilon}(t, du) =\mathbb{E}[M_t-M_{\rho(t)}|\mathcal{F}_{\rho(t)}]=0$ and $\nu(t)\int_{\mathbb{R}}u{\Upsilon}(t, du)=0$, \begin{equation}\label{e3.19'} q(t)=2a +b^2 \widehat{K}^c_t +a^2\nu(t)+b^2\int_{\mathbb{R}}u^2{\Upsilon}(t, du). \end{equation} We define the function $\bar q(t)=\lim_{\rho(s)\downarrow t}q(s)$. It is seen that $\bar q$ is $rd$-continuous and $\bar q(t)=q(\sigma(t))$ if $t$ is right scattered. Since $\{t:\mu(t)>0\}$ is countable and $\operatorname{meas}\{t:\mathbb{E} X_{s,x}^2(t_-)\ne \mathbb{E} X_{s,x}^2(t)\}=0$, \begin{align*} \int_{s}^t{q}(\tau)\mathbb{E} X_{s,x}^2(\tau_-)\nabla\tau &=\int_{(s,t]}{q}(\tau)\mathbb{E} X_{s,x}^2(\tau_-)\,d\tau +\sum_{s<\tau\leq t}q(\tau)\mathbb{E} X_{s,x}^2(\tau_-)\nu(\tau)\\ &=\int_{[s,t)}{\bar q}(\tau)\mathbb{E} X_{s,x}^2(\tau)\,d\tau +\sum_{s\leq\tau< t}q(\sigma(\tau))\mathbb{E} X_{s,x}^2(\tau)\mu(\tau)\\ &=\int_{s}^t\overline{q}(\tau)\mathbb{E} X_{s,x}^2(\tau)\Delta\tau, \end{align*} from which it follows that \begin{align}\label{e2.19'} \mathbb{E} X^2_{s,x}(t)=x^2e_{\overline{q}}(t,s),\quad t\geq s. \end{align} Further, it is known that $$ 0s. $$ Choose $T>0$ such that $\ln \Gamma-\frac{\theta T}2<0$ we obtain $$ \int_s^t\lim_{h\searrow\mu(\tau)}\frac{\ln(1+\overline{q}(\tau)h)}{h} \Delta \tau\leq -\frac{\theta(t-s)}2\quad \forall t>s+T. $$ Thus, the exponential square stability of \eqref{e2.21} implies \begin{align}\label{e2.20bs} \sup\big\{\frac 1{t-s}\int_s^t\lim_{h\searrow\mu(\tau)} \frac{\ln(1+\overline{q}(\tau)h)}{h}\Delta \tau:t>s+T\big\}<0. \end{align} Conversely, supposing that \eqref{e2.20bs} holds, there are $\alpha>0, K^*>0$ such that $0