迭代Brown运动的一个Chung型重对数律

X及Y分别为Rd1及Rd2中的相互独立的标准Brown运动,满足X（0）=Y（0）=0．定义,称为一个迭代Brown运动．本文给出了关于Zd1,d2的一个Chung型重对数律．     

The Compact Law of the Iterated Logarithm for Multivariate Stochastic Approximation Algorithms

Abstract

We consider a sequence (Z _n ) _n≥1 defined by a general multivariate stochastic approximation algorithm and assume that (Z _n ) converges to a solution z* almost surely. We establish the compact law of the iterated logarithm for Z _n by proving that, with probability one, the limit set of the sequence (Z _n  ? z*) suitably normalized is an ellipsoid. We also give the law of the iterated logarithm for the l ^p norms, p ∈ [1, ∞], of (Z _n  ? z*).     

Anticipating Stochastic Differential Equations of Stratonovich Type

We prove the existence and uniqueness of Stratonovich stochastic differential equations where the coefficients and the initial condition may depend on the whole path of the driving Wiener process. Our main hypothesis is that the diffusion coefficient satisfies the Frobenius condition. The solution is given in terms of solutions of ordinary differential equations and the Wiener process. We use this representation to study properties of the solution. Accepted 3 April 1996     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

加权U—统计量的一个重对数律

证明了关于独立同分布随机变量序列的加权Ｕ－统计量的一个重对数律,类似于献「３」证明了一个加权Ｕ－统计量的解耦不等式。     

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.     

Continuity of the Explosion Time in Stochastic Differential Equations

Abstract

Stochastic ordinary differential equations may have solutions that explode in finite time. In this article we prove the continuity of the explosion time with respect to the different parameters appearing in the equation, such as the initial datum, the drift, and the diffusion.     

Extrapolation of the Stochastic Theta Numerical Method for Stochastic Differential Equations

ABSTRACT

The stochastic theta method is a family of implicit Euler methods for approximating solutions to Itô stochastic differential equations. It is proved that the weak error for the stochastic theta numerical method is of the correct form to apply Richardson extrapolation. Several computational examples illustrate the improvement in accuracy of the approximations when applying extrapolation.     

线性过程的强逼近和重对数律

本文讨论由独立同分布随机变量列产生的线性过程的泛函型重对数律和强逼近, 同时又给出由NA随机变量列产生的线性过程的重对数律.     

两参数OU过程的钟重对数律（英）

在本文中，我们证明了两参数OU过程的钟重对数律。     

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.     

关于可交换随机变量序列的重对数律

本文讨论了可交换随机变量序列{Xn:n≥1}的重对数律。     

Multiple Solutions to Stochastic Differential Delay Equations and a Related Comparison Theorem

In this article, we discuss the existence of multiple solutions to a one-dimensional stochastic differential delay equation with continuous drift coefficients and derive a related comparison theorem.     

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.     

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.     

NA序列重对数律的几个极限定理

设{X_n;n≥1}均值为零、方差有限的NA平稳序列。记S_n=∑_(k=1)~n X_k,M_n=maxk≤n|S_k|,n≥1.假设σ~2=EX_1~2+2∑_(k=2)~∞EX_1X_k>0。本文讨论了:当ε 0时,P{M_n≥εσ(2nloglogn)~(1/2)的一类加权级数的精确渐近性质,以及当ε∞时,P{M_n≤εσ(π~2n/(8loglogn))~(1/2)}的一类加权级数的精确渐近性质。这些性质与重对数律和Chung重对数律的速度有关。     

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.     

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.