On Some Properties of Space Inverses of Stochastic Flows

TL;DR

This paper investigates properties of space inverses of stochastic flows generated by jump SDEs, providing moment estimates, a strong limit theorem, and applications to SPDEs, advancing understanding in stochastic analysis.

Contribution

It introduces new moment estimates and limit theorems for jump SDE flows and applies these to establish existence and uniqueness of solutions for certain SPDEs.

Findings

Derived moment estimates for jump SDE flows

Established a strong limit theorem for space inverses

Proved existence and uniqueness of solutions for linear parabolic SPDEs

Abstract

We derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by jump SDEs with adapted coefficients in weighted H\"older norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of continuous SDEs, we derive the existence and uniqueness of classical solutions of linear parabolic second order SPDEs by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case.