Flows for singular stochastic differential equations with unbounded drifts

Abstract

In this paper, we are interested in the following singular stochastic differential equation (SDE)dXt=b(t,Xt)dt+dBt,0≤t≤T, X0=x∈Rd, where the drift coefficient b:[0,T]×Rd⟶Rd is Borel measurable, possibly unbounded and has spatial linear growth. The driving noise Bt is a d− dimensional Brownian motion. The main objective of the paper is to establish the existence and uniqueness of a strong solution and a Sobolev differentiable stochastic flow for the above SDE. Malliavin differentiability of the solution is also obtained (cf. [21,23]). Our results constitute significant extensions to those in [31,30,14,21,23] by allowing the drift b to be unbounded. We employ methods from white-noise analysis and the Malliavin calculus. As application, we prove existence of a unique strong Malliavin differentiable solution to the following stochastic delay differential equationdX(t)=b(X(t−r),X(t,0,(v,η))dt+dB(t),t≥0,(X(0),X0)=(v,η)∈Rd×L2([−r,0],Rd), with the drift coefficient b:Rd×Rd→Rd is a Borel-measurable function bounded in the first argument and has linear growth in the second argument.