SDEs with random and irregular coefficients

Guohuan Zhao

doi:10.4171/rmi/1313

Abstract

We consider It\^o uniformly nondegenerate equations with random coefficients. When the coefficients satisfy some low regularity assumptions with respect to the spatial variables and Malliavin differentiability assumptions on the sample points, the unique solvability of singular SDEs is proved by solving backward stochastic Kolmogorov equations and utilizing a modified Zvonkin type transformation.

Highlights

The main purpose of this work is to study the well-posedness of stochastic differential equations (SDEs) with random and irregular coefficients
In the past half century, a great deal of mathematical effort in stochastic analysis has been devoted to the study of the existence, uniqueness and regularity properties of strong solutions to Itô uniformly nondegenerate stochastic equations with deterministic and irregular drifts
When ∇ ∈ 2 and is bounded, Veretennikov [19] proved the strong existence and uniqueness of solutions to SDE (1.1) by developing the original idea proposed by Zvonkin in [26]

Summary

Introduction

The main purpose of this work is to study the well-posedness of stochastic differential equations (SDEs) with random and irregular coefficients. After adding some Malliavin differentiability conditions on and , they extended the boundedness of time average of a deterministic function depending on a diffusion process with deterministic drift coefficient to random mappings and by investigate the backward stochastic Kolmogorov equation (1.5) ( ≡ I) in some -type space. Inspired by [4] and [26], in this paper we prove a 2+ type estimate (Theorem 3.4) for ( , ), provided that the coefficients satisfy some Malliavin differentiability conditions To achieve this purpose, we first extend the classic Schauder estimate to random PDEs with Banach variables. The main ingredient of this paper is Theorem 3.4, where we give the 2+ estimate for , provided that the Malliavin derivatives of the coefficients satisfy (1.3) To us, such kind of result is new and intriguing.

Schauder Estimates for Random Banach-valued PDEs

Littlewood-Paley decomposition

A basic apriori estimate

Schauder estimate for Backward SPDE

SDEs with random singular coefficients