Strong solution of stochastic differential equations with discontinuous and unbounded coefficients

TL;DR

This paper proves the existence and uniqueness of strong solutions for certain stochastic differential equations with discontinuous and unbounded drift coefficients, extending previous results by combining Zvonkin transformation and Lyapunov methods.

Contribution

It introduces a novel approach that handles discontinuous and unbounded drifts in SDEs, including localized PDE connections and stability results.

Findings

Established local Krylov estimates.

Proved localized stability of solutions.

Extended existence and uniqueness results to more general coefficients.

Abstract

In this paper we study the existence and uniqueness of the strong solution of following d dimensional stochastic differential equation (SDE) driven by Brownian motion: dX(t)=b(t,X(t))dt+a(t,X(t))dB(t), X(0)= x, where B is a d-dimensional standard Brownian motion; the diffusion coefficient a is a Holder continuous and uniformly non-degenerate matrix-valued function and the drift coefficient b may be discontinuous and unbounded, not necessarily in Sobolev space, extending the previous works to discontinuous and unbounded drift coefficient situation. The idea is to combine the Zvonkin transformation with the Lyapunov function approach. To this end, we need to establish a local version of the connection between the solutions of the SDE up to the exit time of a bounded connected open set D and the associated partial differential equation on this domain. As an interesting byproduct, we establish a localized version of the Krylov estimates (Theorem 4.1) and a localized version of the stability result of the stochastic differential equations of discontinuous coefficients (Theorem 4.5).