Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions

TL;DR

This paper establishes a pathwise method for solving one-dimensional stochastic integral equations with non-smooth dispersion and irregular drift, extending classical approaches to broader classes of stochastic equations.

Contribution

It introduces a novel relation between stochastic integral equations and ordinary integral equations, enabling pathwise solutions for equations with non-smooth coefficients.

Findings

Pathwise solutions are obtained for equations with irregular coefficients.

The method generalizes classical stochastic calculus techniques.

Solutions are characterized via an integral equation framework.

Abstract

We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions of certain ordinary integral equations, indexed by a generic element of the underlying probability space. This relation allows us to solve the stochastic integral equations in a pathwise sense.