Successive approximation of infinite dimensional semilinear backward stochastic evolution equations with jumps

In this paper, we study the existence and uniqueness of mild solutions to semilinear backward stochastic evolution equations driven by the cylindrical I$I$-Brownian motion and the Poisson point process in a Hilbert space with non-Lipschitzian coefficients by the successive approximation.     

Stability of semilinear stochastic evolution equations

Stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of Liapunov function. Sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved. Three examples are given to illustrate the theory.     

Existence of solutions of semilinear stochastic delay evolution inclusions in a Hilbert space

In this paper the existence of mild solutions of a class of semilinear stochastic evolution inclusions with delays in a Hilbert space is studied. The results are obtained by using a fixed point theorem for condensing map due to Martelli. The result is illustrated with an example.     

Optimal control for semilinear stochastic evolution equations

Existence, uniqueness, continuous dependence with respect to controls and convergence in the probability of finite differences for controlled semilinear stochastic evolution equations, driven by continuous semimartingales, are considered under Lipschitz and monotone coefficients. The existence of discrete-optimal feedback controls for an associated optimization problem is proved.     

Some results on semilinear stochastic differential equations in hilbert spaces

We study a semilinear stochastic equation by reducing it to a deterministic problem     

A viability theorem of stochastic semilinear evolution equations

A viability theorem of stochastic semilinear evolution equations is discussed under a dissipative condition in terms of uniqueness functions and a stochastic subtangential condition. Our strategy is to interpret a stochastic viability problem into a characterization problem of evolution operators associated with stochastic semilinear evolution equations. The main theorem is a generalization of the results due to Aubin and Da Prato in the case of stochastic differential equations in ℝ ^d .     

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.     

Numerical analysis of semilinear stochastic evolution equations in Banach spaces

The solution of stochastic evolution equations generally relies on numerical computation. Here, usually the main idea is to discretize the SPDE spatially obtaining a system of SDEs that can be solved by e.g., the Euler scheme. In this paper, we investigate the discretization error of semilinear stochastic evolution equations in L^p-spaces, resp. Banach spaces. The space discretization may be done by Galerkin approximation, for the time discretization we consider the implicit Euler, the explicit Euler scheme and the Crank–Nicholson scheme. In the last section, we give some examples, i.e., we consider an SPDEs driven by nuclear Wiener noise approximated by wavelets and delay equation approximated by finite differences.     

Approximate controllability of backward stochastic evolution equations in Hilbert spaces

In this paper, we examine the approximate controllability of a semilinear backward stochastic evolution equations in Hilbert spaces with non-Lipschitz coefficient.     

Harmonic analysis of stochastic equations and backward stochastic differential equations

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in ${\mathcal{R}^p}$ ( ${p\in [1,\infty)}$ ) and backward stochastic differential equations (BSDEs) in ${\mathcal{R}^p\times \mathcal{H}^p}$ ( ${p\in (1, \infty)}$ ) and in ${\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}$ , with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman’s inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear cases for stochastic differential equations (SDEs) and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Hölder inequality for some suitable exponent p ≥ 1. Finally, some relations are established between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite.     

Ergodic backward stochastic difference equations

We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem.     

�������Ͳ��������������ĵ��������΢�ַ���

??A class of backward doubly stochastic differential equations driven by white noises and Poisson random measures are studied in this paper. The definitions of solutions and Yamada-Watanabe type theorem to this equation are established.     

Discretization and simulation of stochastic differential equations

We discuss both pathwise and mean-square convergence of several approximation schemes to stochastic differential equations. We then estimate the corresponding speeds of convergence, the error being either the mean square error or the error induced by the approximation on the value of the expectation of a functional of the solution. We finally give and comment on a few comparative simulation results.     

Linear Volterra backward stochastic integral equations

We present an explicit solution triplet $(Y,Z,K)$ to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process $Y$ is expressed by an integral whose kernel is explicitly given. The processes $Z$ and $K$ are expressed by Hida–Malliavin derivatives involving $Y$.     

超前倒向重随机微分方程

本文研究一类带Lipschitz 系数的超前倒向重随机微分方程。首先利用压缩映像原理得到这类方程的解的存在唯一性,然后给出一维情形下几种不同形式的比较定理,并给出大量的例子来展示所得理论结果的应用。