Strong solutions of some one-dimensional SDEs with random and unbounded drifts

TL;DR

This paper establishes the existence, uniqueness, and differentiability of strong solutions for one-dimensional SDEs with unbounded, random, and deterministic drifts, using probabilistic methods and Malliavin calculus.

Contribution

It introduces new conditions allowing for unbounded, random drifts in SDEs and provides explicit Malliavin derivatives and Sobolev flow properties.

Findings

Proved existence and uniqueness of strong solutions under new drift conditions.

Established Malliavin differentiability and provided explicit derivatives.

Demonstrated the existence of weighted Sobolev differentiable flows.

Abstract

In this paper, we are interested in the following one dimensional forward stochastic differential equation (SDE) \[ d X_{t}=b(t,X_{t},\omega)d t +\sigma d B_{t},\quad 0\leq t\leq T,\quad X_{0}=\,x\in \mathbb{R}, \] where the driving noise $B_{t}$ is a $d$-dimensional Brownian motion. The drift coefficient $b:[0,T] \times\Omega\times \mathbb{R}\longrightarrow \mathbb{R}$ is Borel measurable and can be decomposed into a deterministic and a random part, i.e., $b(t,x,\omega) = b_1(t,x) + b_2(t,x,\omega)$. Assuming that $b_1$ is of spacial linear growth and $b_2$ satisfies some integrability conditions, we obtain the existence and uniqueness of a strong solution. The method we use is purely probabilitic and relies on Malliavin calculus. As byproducts, we obtain Malliavin differentiability of the solutions, provide an explicit representation for the Malliavin derivative and prove existence of weighted Sobolev differentiable flows.