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.     

Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay

In this paper, we prove the existence of mild solutions for a first‐order impulsive semilinear stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay. We consider the cases in which the right hand side is convex or nonconvex valued. The results are obtained by using two different fixed point theorems for multivalued mappings. Copyright © 2015 John Wiley & Sons, Ltd.     

On existence and uniqueness of solutions to uncertain backward stochastic differential equations简

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.     

Constrained LQ Problem with a Random Jump and Application to Portfolio Selection

This paper deals with a constrained stochastic linear-quadratic(LQ for short)optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the It-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations(BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.     

The interrelation between stochastic differential inclusions and set-valued stochastic differential equations

In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L²$L^2$ consisting of square integrable random vectors. We show that for the solution X$X$ to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution x$x$ for this inclusion that is a _‖⋅‖L2${\Vert \cdot \Vert }_{L^2}$-continuous selection of X$X$. This result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions, as well as to consider the viability problem for stochastic differential inclusions.     

由Lévy过程驱动的一类特殊的高维的BSRDE解的存在唯一性

讨论由Brownian运动和Lévy过程共同驱动的线性随机系统的随机LQ问题,其中代价泛函是关于Lévy过程生成的σ-代数取条件期望.得到由Lévy过程驱动的新的多维的倒向随机Riccati方程,利用Bellman拟线性原理和单调收敛方法证明了此随机Riccati方程的解的存在性.     

Optimal variational principle for backward stochastic control systems associated with Lévy processes

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel’s martingales and an independent multi-dimensional Brownian motion,where Teugel’s martin- gales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see e.g.,Nualart and Schoutens’ paper in 2000).We derive the necessary and sufficient conditions for the existence of the op- timal control by means of convex variation methods and duality techniques.As an application,the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (or backward linear-quadratic problem,or BLQ problem for short) is discussed and characterized by a stochastic Hamilton system.     

随机脉冲随机微分方程解的存在唯一性

提出了随机脉冲随机微分方程模型,其中所谓的随机脉冲是指脉冲幅度由随机变量序列驱动,并且脉冲发生的时间也是一个随机变量序列.因此,随机脉冲随机微分方程是对带跳的随机微分方程模型的推广.利用Gronwall不等式、Lipschtiz条件和随机分析技巧,得到了随机脉冲随机微分方程的解的存在唯一性条件.     

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.     

LIMIT THEOREM FOR STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SEMIMARTINGALES IN MANIFOLDS

1IntroductionA limit theorem for the approximation of solutions of stochastic di?erential equations bythose of ordinary di?erential equations was first established by Stroock and Varadhan[8]andits various versions were studied e.g.in[1],[5],[6].They have …     

On Weak Solutions to Stochastic Differential Inclusions Driven by Semimartingales

Abstract

In this paper we consider weak solutions to stochastic inclusions driven by a general semimartingale. We prove the existence of weak solutions and equivalence with the existence of solutions to the martingale problem formulated to such inclusion. Using this we then analyze compactness property of solutions set. Presenting results extend some of those being known for stochastic differential inclusions of Itô's type.     

Existence of Solutions of Nonlinear Neutral Stochastic Differential Inclusions in a Hilbert Space

Abstract

In this paper, the existence of mild solutions of a class of non-linear neutral stochastic differential inclusions in Hilbert space is studied. The results are obtained by using a new fixed point theorem for a condensing map due to Martelli. For an application of the result, the neutral stochastic reaction-diffusion inclusion is also discussed.     

二阶非线性中立型无限时滞随机微分包含的可控性

本文主要在希尔伯特空间中讨论了二阶非线性中立型无限时滞随机微分包含的可控性问题.利用凝聚不动点定理得到了系统可控的一个充分性条件.     

Some properties of stochastic differential equations driven by the G-Brownian motion

In this paper, we study the property of continuous dependence on the parameters of stochastic integrals and solutions of stochastic differential equations driven by the G-Brownian motion. In addition, the uniqueness and comparison theorems for those stochastic differential equations with non-Lipschitz coefficients are obtained.     

Discretizations of Linear Elliptic Partial Differential Inclusions

Differential inclusions provide a suitable framework for modelling choice and uncertainty. In finite dimensions, the theory of ordinary differential inclusions and their numerical approximations is well-developed, whereas little is known for partial differential inclusions, which are the deterministic counterparts of stochastic partial differential equations.

The aim of this article is to analyze strategies for the numerical approximation of the solution set of a linear elliptic partial differential inclusion. The geometry of its solution set is studied, numerical methods are proposed, and error estimates are provided.     

On Comparison Theorem and Solutions of BSDEs for Levy Processes

In this paper, we consider backward stochastic differential equations driven by a Levy process. A comparison theorem and an existence and uniqueness theorem of BSDEs with non-Lipschitz coefficients are obtained.     

BSDEs with a random terminal time driven by a monotone generator and their links with PDEs

In this paper, we study one-dimensional backward stochastic differential equations (BSDE) with a random terminal time driven by a monotone generator, and their links with elliptic partial differential equations. Firstly, we present the case of BSDEs driven by a strictly monotone generator, and next we consider BSDEs driven by a monotone generator.     

一类部分信息的随机控制问题的极值原理

在本文中,我们证明了一类部分信息的随机控制问题的极值原理的一个充分条件和一个必要条件.其中,随机控制问题的控制系统是一个由鞅和Brown运动趋动的随机偏微分方程.     

Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process

In this paper, a new concept of Poisson asymptotically almost automorphy for stochastic processes is introduced. And then, some fundamental properties including composition theorems for the space of such processes are proved. Subsequently, this concept is applied to investigate the existence and uniqueness of asymptotically almost automorphic solutions in distribution to some linear and semilinear stochastic differential equations driven by a Lévy process under some suitable conditions. Finally, an example is given to illustrate the main results.     

The Regularity of Stochastic Convolution Driven by Tempered Fractional Brownian Motion and Its Application to Mean-field Stochastic Differential Equations

In this paper, some properties of a stochastic convolution driven by tempered fractional Brownian motion are obtained. Based on this result, we get the existence and uniqueness of stochastic mean-field equation driven by tempered fractional Brownian motion. Furthermore, combining with the Banach fixed point theorem and the properties of Mittag-Leffler functions, we study the existence and uniqueness of mild solution for a kind of time fractional mean-field stochastic differential equation driven by tempered fractional Brownian motion.