A fixed-point theorem for local operators with applications to stochastic equations

TL;DR

This paper extends fixed-point theorems to local operators in stochastic settings, providing a new tool for analyzing solutions of stochastic differential equations in abstract spaces.

Contribution

It introduces a Schauder-like fixed-point theorem for non-compact, local operators in stochastic spaces, with applications to stochastic differential equations.

Findings

Established a fixed-point theorem for local operators in stochastic spaces.

Demonstrated applications to existence of solutions for stochastic differential equations.

Provided illustrative examples linking the theory to practical stochastic problems.

Abstract

We study weak and strong solutions of nonlinear non-compact operator equations in abstract spaces of adapted random points. The main result of the paper is similar to Schauder's fixed-point theorem for compact operators. The illustrative examples explain how this analysis can be applied to stochastic differential equations.