Stochastic averaging principles for multi-valued stochastic differential equations driven by poisson point Processes

The purpose of this article is to investigate an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by Poisson point processes. The solutions to MSDEs driven by Poisson point processes can be approximated by solutions to averaged MSDEs in the sense of both convergence in mean square and convergence in probability. Finally, an example is presented to illustrate the averaging principle.     

Periodic averaging method for impulsive stochastic differential equations with Lévy noise

The purpose of this paper is to present a periodic averaging method for impulsive stochastic differential equations with Lévy noise under non-Lipschitz condition. It is shown that the solutions of impulsive stochastic differential equations with Lévy noise converge to the solutions of the corresponding averaged stochastic differential equations without impulses     

Averaging principle for multiscale stochastic reaction‐diffusion‐advection equations

Stochastic averaging principle is a powerful tool for studying qualitative analysis of multiscale stochastic dynamical systems. In this paper, we will establish an averaging principle for stochastic reaction‐diffusion‐advection equations with slow and fast time scales. Under suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation.     

Ergodicity and strong limit results for two-time-scale functional stochastic differential equations

This article focuses on a class of two-time-scale functional stochastic differential equations, where the phase space of the segment processes is infinite-dimensional. The systems under consideration have a fast-varying component and a slowly varying one. First, the ergodicity of the fast-varying component is obtained. Then inspired by the Khasminskii’s approach, an averaging principle, in the sense of convergence in the pth moment uniformly in time within a finite time interval, is developed.     

Freidlin-Wentzell's large deviations for stochastic evolution equations

We prove a Freidlin-Wentzell large deviation principle for general stochastic evolution equations with small perturbation multiplicative noises. In particular, our general result can be used to deal with a large class of quasi-linear stochastic partial differential equations, such as stochastic porous medium equations and stochastic reaction-diffusion equations with polynomial growth zero order term and p-Laplacian second order term.     

Large deviations for stochastic nonlinear beam equations

We establish a large deviation principle for the solutions of stochastic partial differential equations for nonlinear vibration of elastic panels (also called stochastic nonlinear beam equations).     

Large deviations and approximations for slow-fast stochastic reaction-diffusion equations

A large deviation principle is derived for a class of stochastic reaction-diffusion partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.     

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This paper concerns stochastic differential equations driven by G-Brownian motion under non-Lipschitz condition which is a much weaker condition with a wider range of applications. Stochastic averaging is established for such non-Lipschitz SDEs where an averaged system is presented to replace the original one in the sense of mean square. An example is presented to illustrate the averaging principle.     

An Averaging Principle for Multivalued Stochastic Differential Equations

The averaging principle for multivalued stochastic differential equations (MSDEs) driven by Brownian motion with Brownian noise is investigated. An averaged MSDEs for the original MSDEs is proposed, and their solutions are quantitatively compared. Under suitable assumptions, it is shown that the solution of the MSDEs converges to that of the original MSDEs in the sense of mean square and also in probability. Two examples are presented to illustrate the averaging principle.     

A transfer principle for multivalued stochastic differential equations

In this paper we prove a transfer principle for multivalued stochastic differential equations.     

On the Application of the Averaging Principle in Stochastic Differential Equations of Hyperbolic Type

We prove a theorem on the application of the Bogolyubov–Mitropol'skii averaging principle to stochastic partial differential equations of the hyperbolic type.     

Dynamic programming in stochastic control of systems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.     

Large deviations for stochastic differential delay equations

In this work, a Freidlin–Wentzell type large deviation principle is established for stochastic differential delay equations. The result in Mohammed and Zhang (2006) [6] is improved.     

Large deviations for stochastic PDE with Lévy noise

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Lévy noise. We use general large deviations theorems of Varadhan and Bryc coupled with the techniques of Feng and Kurtz (2006) [15], viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Lévy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.     

Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.     

The maximum principle for the nonlinear stochastic optimal control problem of switching systems

The aim of this paper is to present a stochastic maximum principle for an optimal control problem of switching systems. It presents necessary conditions of optimality in the form of a maximum principle for stochastic switching systems, in which the dynamic of the constituent processes takes the form of stochastic differential equations. The restrictions on transitions for the system are described through equality constraints.     

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.     

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Stability in distribution of stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems. There are several papers which study this stability for stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching technically. In our paper, we are concerned with the general neutral stochastic functional differential equations with Markovian switching and we derive the sufficient conditions for stability in distribution. At the end of our paper, one example is established to illustrate the theory of our work.     

Properties of set-valued stochastic differential equations

The paper is devoted to properties of set-valued stochastic differential equations. The main result of the paper deals with existence and uniqueness of solutions. Furthermore, a connection between solutions of stochastic differential inclusions and solutions of set-valued stochastic differential equations are given. The result of the paper extends a lot of particular results dealing with such type equations.     

Some result on stochastic partial differential equations by the stochastic characteristics method

We study a class of second order (in the drift term) stochastic partial differential equations by the stochastic characteristics method, as developped by Kunita for the first order stochastic partial differential equations. With this method the original problem is transformed in a family of deterministic parabolic problems.