Exponential stability for stochastic neutral partial functional differential equations

In this paper, we consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139-154], Liu and Truman [K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50 (2000) 273-278] and Taniguchi [T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl. 16 (1998) 965-975; T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53 (1995) 41-52] are generalized and improved.     

A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations

Abstract

In many cases, the existence and uniqueness of the solution of a differential equation are proved using fixed point theory. In this paper, we utilize the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations. We first discuss some properties of a class of Volterra integral operators and then establish a new generalized Gronwall integral inequality with a double singularity. Finally, we use the properties and integral inequality to study the well-posedness of mild solution to the semilinear fractional stochastic differential equations. One sees that it is concise and effectiveness using the previous results to investigate the well-posedness of the mild solution.     

Existence results of semilinear differential variational inequalities without compactness

ABSTRACT

The purpose of this paper is to study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing semilinear evolution equations and generalized variational inequalities. First we prove essential properties of the solution set for generalized variational inequalities. Then without requiring any compactness condition for the evolution operator or for the nonlinear term, two existence results for mild solutions are established by applying a weak topology technique combined with a fixed point theorem.     

Existence of solutions for nonlinear fractional stochastic differential equations

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory.     

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

Stability of Mild Solutions of Stochastic Evolution Equations with Variable Delay

Abstract

In this paper, we consider the existence and stability problems associated with semilinear stochastic evolution equations with variable delay in infinite dimensions. To be precise, we first study an existence result and then the exponential stability of a mild solution as well as asymptotic stability in probability of its sample paths. Such results are established employing a comparison principle under less restrictive hypothesis than the Lipschitz condition on the nonlinear terms. An application is included to illustrate the theory.     

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called "Acquistapace–Terreni" conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.     

Perturbation method for nonlocal impulsive evolution equations

This paper deals with the existence of mild solutions for a class of semilinear nonlocal impulsive evolution equations in ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear nonlocal impulsive evolution equation is established. With the aid of the theorem, the existence of mild solutions for nonlinear nonlocal impulsive evolution equation is obtained by using perturbation method and monotone iterative technique. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. Moreover, we present two examples to illustrate the feasibility of our abstract results.     

On stability for a class of semilinear stochastic evolution equations

Sufficient conditions for almost surely asymptotic stability with a certain decay function of sample paths, which are given by mild solutions to a class of semilinear stochastic evolution equations, are presented. The analysis is based on introducing approximating system with strong solution and using a limiting argument to pass on some properties of strong solution to our purposes. Several examples are studied to illustrate our theory. In particular, by means of the derived results we lose conditions of certain stochastic evolution systems from Haussmann (1978) to obtain the pathwise stability for mild solution with probability one.     

Asymptotic stability of solutions for some classes of impulsive differential equations with distributed delay

In this paper we show the asymptotic stability of the solutions of some differential equations with delay and subject to impulses. After proving the existence of mild solutions on the half-line, we give a Gronwall–Bellman-type theorem. These results are prodromes of the theorem on the asymptotic stability of the mild solutions to a semilinear differential equation with functional delay and impulses in Banach spaces and of its application to a parametric differential equation driving a population dynamics model.     

Feedback control systems governed by evolution equations

ABSTRACT

The goal of this paper is to provide systematic approaches to study the feedback control systems governed by evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray–Schauder alternative fixed point theorem with Lipschitz conditions and some types of boundedness conditions. Next, by using the Filippove theorem and the Cesari property, a new set of sufficient assumptions are formulated to guarantee the existence of feasible pairs for the feedback control systems. Some existence results for an optimal control problem are given. Finally, we apply our main result to obtain a controllability result for semilinear evolution equations and existence results for a class of differential variational inequalities and Clarke's subdifferential inclusions.     

Doubly reflected BSDEs driven by a Lévy process

In this paper, we show the existence and uniqueness of the solution for a class of doubly reflected backward stochastic differential equations driven by a Lévy process (DRBSDELs in short) by means of the penalization method as well as the fixed point theorem. In addition, we obtain the comparison theorem for the solutions of DRBSDELs. As an application, we give a probabilistic formula for the viscosity solution of a class of partial differential-integral equations (PDIEs in short) with two obstacles.     

Reflected BSDE's with discontinuous barrier and application

We deal with reflected backward stochastic differential equations with right continuous and left limited barrier. We show the existence and uniqueness of the solution and we give a comparison theorem. As an application, we study the link between such an equations with stochastic mixed control problems.     

The density evolution of the killed McKean–Vlasov process

ABSTRACT

The density evolution of McKean–Vlasov stochastic differential equations in the presence of an absorbing boundary is analysed where the solution to such equations corresponds to the dynamics of partially killed large populations. By using a fixed point theorem, we show that the density evolution is characterized as the solution of an integro-differential Fokker–Planck equation with Cauchy–Dirichlet data. This problem arises naturally within mean field game theory.     

EXISTENCE AND STABILITY OF SOLUTIONS OF STOCHASTIC SEMILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study stochastic semilinear functional differential equations in a Hilbert space. First, we prove the existence and uniqueness of a mild solution under two sets of hypotheses. We then consider the exponential stability of the second moment of the solution process of such equations as well as the exponential stability and asymptotic stability in probability of its sample paths. We further consider global stability in the mean. Such results are obtained using both local Lipschitz and non-Lipschitz nonlinearities. Our method is an interplay of the method of successive approximations and a comparison principle. Two applications are included to motivate this study.     

Semigroups of locally Lipschitz operators associated with semilinear evolution equations

In this paper we introduce the notion of semigroups of locally Lipschitz operators which provide us with mild solutions to the Cauchy problem for semilinear evolution equations, and characterize such semigroups of locally Lipschitz operators. This notion of the semigroups is derived from the well-posedness concept of the initial-boundary value problem for differential equations whose solution operators are not quasi-contractive even in a local sense but locally Lipschitz continuous with respect to their initial data. The result obtained is applied to the initial-boundary value problem for the complex Ginzburg–Landau equation.     

On the support of solutions to stochastic differential equations with path-dependent coefficients

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron–Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock–Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on functional Itô calculus.     

带跳的分数倒向重随机微分方程及相应的随机积分偏微分方程

本文首次把Poisson随机测度引入分数倒向重随机微分方程,基于可料的Girsanov变换证明由Brown运动、Poisson随机测度和Hurst参数在(1/2,1)范围内的分数Brown运动共同驱动的半线性倒向重随机微分方程解的存在唯一性.在此基础上,本文定义一类半线性随机积分偏微分方程的随机黏性解,并证明该黏性解由带跳分数倒向重随机微分方程的解唯一地给出,对经典的黏性解理论作出有益的补充.     

Approximate controllability of impulsive semilinear stochastic system with delay in state

Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process that can be modeled by impulsive differential equations. This article studies the approximate controllability of impulsive semilinear stochastic system with delay in state in Hilbert spaces. Assuming the conditions for the approximate controllability of the corresponding deterministic linear system, we obtain the sufficient conditions for the approximate controllability of the impulsive semilinear stochastic system with delay in state. The results are obtained by using Banach fixed point theorem. Finally, two examples are given to illustrate the developed theory.     

Generalized Directional Gradients, Backward Stochastic Differential Equations and Mild Solutions of Semilinear Parabolic Equations

We study a forward-backward system of stochastic differential equations in an infinite-dimensional framework and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows us to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions. The use of the generalized directional gradient allows us to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction--diffusion equations), where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black--Scholes or Hamilton-Jacobi-Bellman type.