On Weak Solutions of Stochastic Differential Equations

A new proof of existence of weak solutions to stochastic differential equations with continuous coefficients based on ideas from infinite-dimensional stochastic analysis is presented. The proof is fairly elementary, in particular, neither theorems on representation of martingales by stochastic integrals nor results on almost sure representation for tight sequences of random variables are needed.     

On Weak Solutions of Stochastic Differential Equations II

In the first part of this article a new method of proving existence of weak solutions to stochastic differential equations with continuous coefficients having at most linear growth was developed. In this second part, we show that the same method may be used even if the linear growth hypothesis is replaced with a suitable Lyapunov condition.     

随机微分方程终值问题的弱解(英)

设｛Wt．Ft．t∈［0．T］｝为概率空间（Ω，P）上的标准α维Brown运动，为由它生成的自然σ－代数流．本文讨论了如下随机微分方程终值问题弱解的存在性：其中ξ∈L2（Ω，P；Rn），g：［0，T」×Rn×Rnd→Rn为有界可测函数．此外，还讨论了它在金融市场期权定价问题中的应用．     

Weak Solutions for SPDEs and Backward Doubly Stochastic Differential Equations

We give the probabilistic interpretation of the solutions in Sobolev spaces of parabolic semilinear stochastic PDEs in terms of Backward Doubly Stochastic Differential Equations. This is a generalization of the Feynman–Kac formula. We also discuss linear stochastic PDEs in which the terminal value and the coefficients are distributions.     

随机微分方程弱解的稳定性和存在性

本文研究了驱动项为无穷维Ｂｒｏｗｎ运动的一般Ｉｔ随机微分方程，给出了还问题的解和弱解的存在性关系，证明了在线性增长条件下，方程弱解的稳定性和存在性定理．     

L p Solutions of One-Dimensional Backward Stochastic Differential Equations with Continuous Coefficients

In this article, we study one-dimensional backward stochastic differential equations with continuous coefficients. We show that if the generator f is uniformly continuous in (y, z), uniformly with respect to (t, ω), and if the terminal value ξ ∈L ^p (Ω, ? _T , P) with 1 < p ≤ 2, the backward stochastic differential equation has a unique L ^p 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.     

Integrability of Solutions to Mixed Stochastic Differential Equations

We prove that the standard conditions for the unique solvability of a mixed stochastic differential equation guarantee that its solution possesses finite moments. We also give conditions supplying the existence of exponential moments. For a special equation whose coefficients do not satisfy the linear growth condition, we prove the integrability of its solution.     

On Convergence of One-Step Schemes for Weak Solutions of Quantum Stochastic Differential Equations

Several one-step schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with creation, annihilation and gauge operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. It is shown that the Euler–Maruyama scheme,with respect to weak convergence criteria for Itô stochastic differential equation is a special case of Euler schemes in this framework. Numerical examples are given.     

随机微分方程的强解

在这篇文章中我们通过一种去掉扩散系数的变换证明了随机微分方程强解的存在唯一性.     

Global Weak Solutions to One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity Coefficients

We prove the global existence of weak solutions of the one-dimensional compressible Navier-stokes equations with density-dependent viscosity. In particular, we assume that the initial density belongs to L^1 and L^∞, module constant states at x = -∞ and x = ＋∞, which may be different. The initial vacuum is permitted in this paper and the results may apply to the one-dimensional Saint-Venant model for shallow water.     

倒向随机微分方程弱解（英）

引入倒向随机微分方程弱解的概念，应用Girsanov变换，建立了两类倒向随机微分方程（0．1）和（0．2）弱解存在的等价性，由此得到倒向 随机微分方程弱解存在的几个充分条件。     

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.     

Topological Solutions of Noncommutative Stochastic Differential Equations

Abstract

A general approach is introduced to studying the properties of solutions of an arbitrary noncommutative stochastic differential equation (NSDE) in several interesting locally convex operator topologies, grouped into two main sets comprising the strong/λ^?-topologies and the weak topologies. Results concerning the existence and uniqueness of solutions in these topologies are established. The approach is based on two reformulations of the NSDE, corresponding to the two sets of topologies, and is well-suited for characterizing, both analytically and numerically, various topological features of the solutions of an NSDE.     

Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D _t u _t  = D _x u _x  + (X _u  + λ₀(u)u _t  + λ₁(u)u _x )[Wdot] where X is a continuous vector field on M, λ₀ and λ₁ are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ? with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.     

Strong Markov Continuous Local Martingales and Solutions of One-Dimensional Stochastic Differential Equations (Part III)

This last part of the present paper is devoted to one-dimensional stochastic differential equations driven by a WIENER process. In Section 4, we give a survey on existence, uniqueness, and various other aspects of solutions. In Section 5, which was the starting point of the present paper, we describe the connection between strong MARKOV continuous local martingales and solutions of one-dimensional stochastic differential equations without drift.