Construction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada-Watanabe principle

TL;DR

This paper develops a method to construct unique strong solutions for SDEs that are Malliavin differentiable, using a compactness criterion, without relying on the Yamada-Watanabe principle, and applies this to derive a Bismut-Elworthy-Li formula.

Contribution

It introduces a novel approach employing a compactness criterion to construct Malliavin differentiable solutions under integrability conditions, bypassing the Yamada-Watanabe principle.

Findings

Constructed unique strong solutions that are Malliavin differentiable.

Derived a Bismut-Elworthy-Li formula for Kolmogorov equations.

Established the applicability of the method under integrability conditions.

Abstract

In this paper we aim at employing a compactness criterion of Da Prato, Malliavin, Nualart for square integrable Brownian functionals to construct unique strong solutions of SDE's under an integrability condition on the drift coefficient. The obtained solutions turn out to be Malliavin differentiable and are used to derive a Bismut-Elworthy-Li formula for solutions of the Kolmogorov equation.