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−→Rdis Borel measurable, possibly unbounded and has spatial linear growth. The driving noise Btis 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 bto 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 equation.