A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.     

Skorohod problem and multivalued stochastic evolution equations in Banach spaces

By solving a deterministic Skorohod problem in the framework of evolutional triple, we prove the existence and uniqueness of solutions to multivalued stochastic evolution equations involving maximal monotone operators. The existence and uniqueness of invariant measures associated with the solutions as Markov processes are also considered in the present paper. Moreover, we apply the results to stochastic differential equations with normal reflecting boundary conditions and with singular drift terms, as well as a class of multivalued nonlinear stochastic partial differential equations with possibly discontinuous coefficients.     

Existence of solutions for nonlinear fractional stochastic differential equations

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory.     

Strong Solutions of Semilinear Parabolic Equations with Measure Data and Generalized Backward Stochastic Differential Equations

We prove that under natural assumptions on the data strong solutions in Sobolev spaces of semilinear parabolic equations in divergence form involving measure on the right-hand side may be represented by solutions of some generalized backward stochastic differential equations. As an application we provide stochastic representation of strong solutions of the obstacle problem by means of solutions of some reflected backward stochastic differential equations. To prove the latter result we use a stochastic homographic approximation for solutions of the reflected backward equation. The approximation may be viewed as a stochastic analogue of the homographic approximation for solutions to the obstacle problem.     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity 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.     

关于拟线性混合型边界问题的概率表示

关于某些抛物型和椭圆型偏微分方程的混合边界问题的解被表示为一类联系于Ito正向反射边界随机微分方程的反向随机微分方程的解.     

On a class of backward doubly stochastic differential equations

In this paper, a new class of backward doubly stochastic differential equations is studied. This type of equations has a more general form of the forward Itô integrals compared to the ones which have been studied until now. We conclude that unique solutions of these equations can be represented with the help of solutions of the corresponding backward doubly stochastic differential equations, considered earlier in paper [5] by Pardoux and Peng. Some comparison theorems are also given, as well as a probabilistic interpretation for solutions of the corresponding quasilinear stochastic partial differential equations.     

具有非Lipschitz和非增长条件的带跳倒向随机微分方程

本文研究一类带Poisson跳的倒向随机微分方程。在方程的系数满足非增长条件和非Lipschitz条件下,讨论方程适应解的存在唯一性和稳定性。为了证明解的存在性,首先通过函数变换,构造出一逼近序列,然后运用推广的Bihari不等式和Lebesgue控制收敛定理证明该逼近序列是收敛的,得到逼近序列的极限就是方程的适应解。解的唯一性和稳定性主要运用了Bihari不等式和推广的Bihari不等式来进行证明。     

一类具有无穷时滞中立型非稠定脉冲随机泛函微分方程积分解的存在性

本文讨论了一类具有无穷时滞中立型非稠定脉冲随机泛函微分方程,利用Sadovskii不动点原理等工具得到了其积分解的存在性,给出其在一类二阶无穷时滞中立型非稠定脉冲随机偏微分方程积分解的存在性中的应用.     

Ergodic BSDEs under weak dissipative assumptions

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in Fuhrman et al. (2009) [12]. In other words we do not need to require the uniform exponential decay of the difference of two solutions of the underlying forward equation, which, on the contrary, is assumed to be non-degenerate.We show the existence of solutions by the use of coupling estimates for a non-degenerate forward stochastic differential equation with bounded measurable nonlinearity. Moreover we prove the uniqueness of “Markovian” solutions by exploiting the recurrence of the same class of forward equations.Applications are then given for the optimal ergodic control of stochastic partial differential equations and to the associated ergodic Hamilton-Jacobi-Bellman equations.     

A Stochastic Maximum Principle for a Stochastic Differential Game of a Mean-Field Type

We construct a stochastic maximum principle (SMP) which provides necessary conditions for the existence of Nash equilibria in a certain form of N-agent stochastic differential game (SDG) of a mean-field type. The information structure considered for the SDG is of a possible asymmetric and partial type. To prove our SMP we take an approach based on spike-variations and adjoint representation techniques, analogous to that of S.?Peng (SIAM J. Control Optim. 28(4):966?C979, 1990) in the optimal stochastic control context. In our proof we apply adjoint representation procedures at three points. The first-order adjoint processes are defined as solutions to certain mean-field backward stochastic differential equations, and second-order adjoint processes of a first type are defined as solutions to certain backward stochastic differential equations. Second-order adjoint processes of a second type are defined as solutions of certain backward stochastic equations of a type that we introduce in this paper, and which we term conditional mean-field backward stochastic differential equations. From the resulting representations, we show that the terms relating to these second-order adjoint processes of the second type are of an order such that they do not appear in our final SMP equations. A?comparable situation exists in an article by R.?Buckdahn, B.?Djehiche, and J.?Li (Appl. Math. Optim. 64(2):197?C216, 2011) that constructs a SMP for a mean-field type optimal stochastic control problem; however, the approach we take of using these second-order adjoint processes of a second type to deal with the type of terms that we refer to as the second form of quadratic-type terms represents an alternative to a development, to our setting, of the approach used in their article for their analogous type of term.     

On solutions set of a multivalued stochastic differential equation

We analyse multivalued stochastic differential equations driven by semimartingales. Such equations are understood as the corresponding multivalued stochastic integral equations. Under suitable conditions, it is shown that the considered multivalued stochastic differential equation admits at least one solution. Then we prove that the set of all solutions is closed and bounded.     

Stochastic evolution equations in Hilbert spaces

In this paper we consider Ito's stochastic differential equation in Hilbert spaces. A strong solution is generated by the difference approximation. A regularity result is obtained for solutions to a class of parabolic stochastic partial differential equations. Hyperbolic stochastic evolution equations are also discussed.     

On anticipated backward stochastic differential equations with Markov chain noise

In 2013, Lu and Ren considered anticipated backward stochastic differential equations driven by finite state, continuous time Markov chain noise and established the existence and uniqueness of the solutions of these equations and a scalar comparison theorem. In this article, we provide an estimate for their solutions and study the duality between these equations and stochastic differential delayed equations with Markov chain noise. Finally, we derive another comparison theorem for these solutions depending only on the two drivers.     

Smooth solutions of non-linear stochastic partial differential equations driven by multiplicative noises

In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.     

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.     

On asymptotic stability of solutions of stochastic differential equations with constant drift

In this paper we provide conditions for asymptotic stability of solutions of stochastic differential equations with constant drift. The main result of the paper extends the results of Khasminskii and Nevelson ([5]), where only the stochastic differential equations with recurrent solutions were studied     

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.     

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.