Finite dimensional filter systems in discrete time

Under regularity conditions, a finite dimensional filter system exists for a partially observable process if and only if the conditional distributions involved each form an exponential family of distributions. The filter equation can be derived directly from the exponential representations of these families.     

与多值随机微分方程相关马氏半群的无穷小生成元（英文）

当多值随机微分方程的扩散及漂移系数满足利普希兹连续性条件时,我们考虑其解的无穷小生成元问题.为了找出该无穷小生成元的核,我们研究了对应的多值椭圆方程及其粘性解.     

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Under the condition that the coefficients are Lipschitz continuous, we study the infinitesimal generator of Markov semigroup corresponding to the multivalued stochastic equation. In order to provide a core of the infinitesimal generator, we investigate the associated multivalued elliptic equation and its viscosity solutions.     

Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin''s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.     

随机偏微分方程——建模,分析与有效动力学

综述随机偏微分方程的基本概念、理论、方法与应用,内容包括Hilbert空间中的Wiener过程、Ito随机积分、随机偏微分方程的解及其有效动力学。还介绍了随机偏微分方程的粗糙轨道、正则结构以及在Kardar-ParisiZhang(KPZ)方程中的应用。还介绍了段金桥与王伟的著作《Effective Dynamics of Stochastic Partial Differential Equations(随机偏微分方程的有效动力学)》的基本内容。     

共振情形下次线性椭圆偏微分方程的解

研究了共振下的微分方程Δｕ＋λ_１ｕ＋ｇ（ｘ，ｕ）＝０，ｘ∈?Ω；ｕ｜_?Ω＝０．在ｇ（ｘ，ｕ）关于ｕ次线性的情形，证明了解的存在性，从而部分地回答了Ｆｉｇｕｅｉｒｅｄｏ ａｎｄ Ｍａｓｓａｂｉ的一个问题．     

Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

We consider a second order semi-elliptic differential operator L with measurable coefficients, in divergence form, and the semilinear parabolic system of PDEs

A Note on the Solution of a Stochastic Partial Differential Equation

Abstract

A physical model is described which justifies the appearance of a stochastic term in the two-dimensional Navier–Stokes equations. In this model, a linear oppositional control term accrues as well. The resulting stochastic partial differential equation is shown to have a unique stationary solution.     

一类椭圆型随机偏微分方程弱解的存在性

设$D$是$R^N$ ($N>1$)中有界开集,$(\Omega, {\cal F}, P)$是一个完备的概率空间.该文研究了下列随机边值问题弱解的存在性问题\[\left\{\begin{array}{ll}-{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &;(x,\omega)\in D\times \Omega,\\u=0, &;(x,\omega)\in \partial D\times \Omega,\end{array}\right.\]其中, div与 $\nabla $ 表示仅对 $x$求微分. 首先,作者引入了弱解的概念; 然后,作者转化随机问题为高维确定性问题;最后,作者证明了该问题弱解的存在性.     

On a Stochastic Partial Differential Equation with Non-local Diffusion

In this paper, we prove existence, uniqueness and regularity for a class of stochastic partial differential equations with a fractional Laplacian driven by a space-time white noise in dimension one. The equation we consider may also include a reaction term.      

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II

We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of L^p convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.     

Backward Stochastic Differential Equation with Two Reflecting Barriers and Jumps

Abstract

In this paper, by using a penalization as well as a fixed point methods, we prove existence and uniqueness of the solution for the one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process.     

A Three-Field Finite Element Method for Elliptic Partial Differential Equations Driven by Stochastic Loads

Abstract

This paper is concerned with the application of nonconforming finite element methods to stochastic partial differential equations. We present a mixed formulation of a three-field finite element method applied to an elliptic model problem involving stochastic loads. We then derive the exact form for the expected value and variance of the solution. Additionally, the rate of convergence for the stochastic error is presented. Finally, we demonstrate through numerical experiments that the method is robust and reliable.     

Space Semi-Discretisations for a Stochastic Wave Equation

We study an approximation scheme for a nonlinear stochastic wave equation in one-dimensional space, driven by a spacetime white noise. The sequence of approximations is obtained by discretisation of the Laplacian operator. We prove L ^p -convergence to the solution of the equation and determine the rate of convergence. As a corollary, almost sure convergence, uniformly in time and space, is also obtained. Finally, the speed of convergence is tested numerically.⋆Supported by the grant BMF 2003-01345 from the Dirección General de Investigación, Ministerio de Ciencia y Tecnología, Spain.     

Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

Stochastic Theta Method for a Reflected Stochastic Differential Equation

In this article, a stochastic theta method for a reflected stochastic differential equation is proposed. When the parameter θ = 0, this method coincides with the projection Euler scheme; while when the parameter θ = 1, it is called an implicit projection Euler scheme which is first proposed in this article. Under some conditions, the strong convergence and the A-stability of this numerical scheme are proved.     

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White Noise I

We approximate quasi-linear parabolic SPDEs substituting the derivatives in the space variable with finite differences. When the nonlinear terms in the equation are Lipschitz continuous we estimate the rate of L^p convergence of the approximations and we also prove their almost sure uniform convergence to the solution. When the nonlinear terms are not Lipschitz continuous we obtain this convergence in probability, if the pathwise uniqueness for the equation holds.     

Anticipative Stochastic Differential Equations with Non-smooth Diffusion Coefficient

In this paper we prove the existence and uniqueness of the solutions to the one-dimensional linear stochastic differential equation with Skorohod integral Xt（ω）=η（w）＋∫^t 0 asXs（ω）dWs＋∫^t 0 bsXs（ω）ds, t∈[0,1] where （Ws） is the canonical Wiener process defined on the standard Wiener space （W,H,u）, a is non-smooth and adapted, but η and b may be anticipating to the filtration generated by （Ws）. The intention of the paper is to eliminate the regularity of the diffusion coefficient a in the Malliavin sense, in the existing literature. The idea is to approach the non-smooth diffusion coefficient a by smooth ones.