Solutions Of One-dimensional Stochastic Equations Research Articles

On Multidimensional SDEs Without Drift and with A Time-Dependent Diffusion Matrix

Abstract We study multidimensional stochastic equations where x o is an arbitrary initial state, W is a d-dimensional Wiener process and is a measurable diffusion coefficient. We give sufficient conditions for the existence of weak solutions. Our main result generalizes some results obtained by A. Rozkosz and L. Słomiński [Stochastics Stochasties Rep. 42: 199–208, 1993] and T. Senf [Stochastics Stochastics Rep. 43: 199–220, 1993] for the existence of weak solutions of one-dimensional stochastic equations and also some results by A. Rozkosz and L. Słomiński [Stochastic Process. Appl. 37: 187–197, 1991], [Stochastic Process. Appl. 68: 285–302, 1997] for multidimensional equations. Finally, we also discuss the homogeneous case.

Convergence of Solutions of One-Dimensional Stochastic Equations

One-dimensional stochastic equations are considered whose coefficients depend on a small parameter in an irregular way. In terms of coefficients, necessary and sufficient conditions are obtained for the weak convergence of their solutions to the solution of the stochastic equation. Examples are given.

On Strong and Weak Solutions of One-Dimensional Stochastic Equations with Boundary Conditions

On Strong and Weak Solutions of One-Dimensional Stochastic Equations with Boundary Conditions