Strong Feller property and continuous dependence on initial data for one-dimensional stochastic differential equations with Hölder continuous coefficients

Hua Zhang

doi:10.1214/20-ecp284

Abstract

In this paper, under the assumption of Hölder continuous coefficients, we prove the strong Feller property and continuous dependence on initial data for the solution to one-dimensional stochastic differential equations whose proof are based on the technique of local time, coupling method and Girsanov’s transform.

Highlights

In the present paper, we are concerned with the problem of the strong Feller property and continuous dependence on initial data for the following one-dimensional stochastic differential equations (SDEs in short) with Hölder continuous coefficients: t tX(t, x) = x + b(X(s, x))ds + σ(X(s, x))dW (s), (1.1)where W (t) is a standard Brownian motion on a filtered probability space (Ω, F, P; (Ft)t≥0), b : R → R and σ : R → R are two continuous functions.As far as we know, there are many ways to establish the strong Feller property on Ptφ(x) := E[φ(X(t, x))]
Under the assumption of Hölder continuous coefficients, we prove the strong Feller property and continuous dependence on initial data for the solution to one-dimensional stochastic differential equations whose proof are based on the technique of local time, coupling method and Girsanov’s transform
We are concerned with the problem of the strong Feller property and continuous dependence on initial data for the following one-dimensional stochastic differential equations (SDEs in short) with Hölder continuous coefficients: t t

Summary

Introduction

We are concerned with the problem of the strong Feller property and continuous dependence on initial data for the following one-dimensional stochastic differential equations (SDEs in short) with Hölder continuous coefficients:. For the case of non-Lipschitz continuous coefficient, Zhang (cf [8]) first used the coupling method combined with Girsanov’s transformation to establish the strong Feller property for the solution to Eq (1.1) with the Log-Lipschitz continuous coefficients. We prove the strong Feller property for SDEs with Hölder continuous coefficients whose proof is based on the technique of local time, coupling method and Girsanov’s transform. Using a precise estimate on local times, the continuous dependence on initial data for SDEs with Hölder continuous coefficients is established, which was not obtained before. When we do not want to emphasis this dependence we just use C instead

Preliminaries

Strong Feller property

Continuous dependence on initial data