Flows for Singular Stochastic Differential Equations with Unbounded Drifts

TL;DR

This paper proves existence, uniqueness, and differentiability of solutions for singular stochastic differential equations with unbounded drifts, extending previous results and applying white-noise analysis and Malliavin calculus.

Contribution

It establishes strong solutions and stochastic flows for SDEs with unbounded drifts, including applications to stochastic delay differential equations, which was not previously achieved.

Findings

Existence and uniqueness of strong solutions for unbounded drifts.

Sobolev differentiable stochastic flows are constructed.

Solutions are Malliavin differentiable, enabling further analysis.

Abstract

In this paper, we are interested in the following singular stochastic differential equation (SDE) $${\rm d} X_t = b(t,X_t) {\rm d} t + {\rm d} B_{t},\ 0\leq t\leq T,\ X_0 = x \in \mathbb{R}^d,$$ where the drift coefficient $b:[0,T]\times \mathbb{R}^{d}\longrightarrow \mathbb{R}^{d}$ is Borel measurable, possibly unbounded and has spatial linear growth. The driving noise $B_{t}$ 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.\cite{MMNPZ13, MNP2015}). Our results constitute significant extensions to those in \cite{Zvon74, Ver79, KR05, MMNPZ13, MNP2015} 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 equation $${\rm d} X (t) = b (X(t-r), X(t,0,(v,\eta)) {\rm d} t + {\rm d} B(t), \,t \geq 0 ,\textbf{ } (X(0), X_0)= (v, \eta) \in \mathbb{R}^d \times L^2 ([-r,0], \mathbb{R}^d),$$ with the drift coefficient $b: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Borel-measurable function bounded in the first argument and has linear growth in the second argument.