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.     

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 Partial Differential Equations on Manifolds~II. Nonlinear Filtering

The conditional law of an unobservable component x(t) of a diffusion (x(t),y(t)) given the observations {y(s):s[0,t]} is investigated when x(t) lives on a submanifold of . The existence of the conditional density with respect to a given measure on is shown under fairly general conditions, and the analytical properties of this density are characterized in terms of the Sobolev spaces used in the first part of this series.     

Maximum Principle for Stochastic Differential Games with Partial Information

In this paper, we first deal with the problem of optimal control for zero-sum stochastic differential games. We give a necessary and sufficient maximum principle for that problem with partial information. Then, we use the result to solve a problem in finance. Finally, we extend our approach to general stochastic games (nonzero-sum), and obtain an equilibrium point of such game.     

Quasi-sure Limit Theorem of Parabolic Stochastic Partial Differential Equations

In this paper we prove a quasi-sure limit theorem of parabolic stochastic partial differentialequations with smooth coefficients and some initial conditions,by the way,we obtain the quasi-surecontinuity of the solution.     

概率方法解拟线性偏微分方程

本文用随机分析方法证明了拟线性抛物型方程ｕｔ＋ｆ（ｕ）ｕｘ、ｕｘｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）在ｕ０有界可测，ｆ连续且ｆ＞０条件下，其解当→０时收敛于拟线性方程ｕｔ＋ｆ（ｕ）ｕｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）的熵解，即论证了“沾性消失法”解此方程的正确性，１９５７年Ｏｌｅｉｎｉｋ曾用差分方法解决了此问题。这里用概率方法重新获得此结果。     

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 Doubly Stochastic Differential Equations with Jumps and Stochastic Partial Differential-Integral Equations

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.     

Quasi-sure Limit Theorem of Parabolic Stochastic Partial Differential Equations

In this paper we prove a quasi-sure limit theorem of parabolic stochastic partial differential equations with smooth coefficients and some initial conditions, by the way, we obtain the quasi-sure continuity of the solution.     

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

This paper is concerned with the numerical approximations of semi-linear stochastic partial differential equations of elliptic type in multi-dimensions. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method. Numerical results demonstrate the good performance of the spectral method.     

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。     

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.     

比较原理和带马尔可夫调制的随机泛函微分方程

本文考虑带马尔可夫调制的随机泛函微分方程解的不稳定性,通过建立的新的比较原理,得到一些不稳定的判据.     

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.     

Impulsive Noise Driven One-Dimensional Higher-Order Fractional Partial Differential Equations

We are going to study a kind of stochastic fractional partial differential equation driven by an impulsive noise, which is singular not only in time but also in space. We first study the existence and uniqueness of the solution and then investigate the regularity of the solution in its spatial variable which depends on the order of the fractional operator, and deeply relies on the precise analysis of the kernel generated by our operator. In addition, we also discuss the stochastic flow property of the solutions.     

Existence and Uniqueness of Solutions to Time-delays Stochastic Fractional Differential Equations with Non-Lipschitz Coefficients

In this paper, we consider the existence and uniqueness of solutions to time-varying delays stochastic fractional differential equations (SFDEs) with non-Lipschitz coefficients. By using fractional calculus and stochastic analysis, we can obtain the existence result of solutions for stochastic fractional differential equations.     

分式噪声驱动的一类随机偏微分方程的非参数估计}

研究了一类由分式噪声所驱动的随机偏微分方程的统计推断. 先构造了偏微分算子时间 相依系数的非参数估计量, 然后得到了该估计在最大值范数下的收敛率和渐近正态性. 该收敛率 由系数的平滑参数和分式噪声的Hurst参数共同决定.     

Approximate Controllability of Fractional Impulsive Partial Neutral Stochastic Differential Inclusions with State-Dependent Delay and Fractional Sectorial Operators

In this article, the approximate controllability of fractional impulsive partial neutral stochastic differential inclusions with state-dependent delay and fractional sectorial operators in Hilbert spaces is studied. By using the stochastic analysis, the fractional sectorial operators and a fixed point theorem for multi-valued maps combined with approximation techniques, we discuss a new set of su?cient conditions for the approximate controllability of the systems under the mixed Lipschitz and Carathéodory conditions. An example is provided to illustrate the obtained theory.