Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

TL;DR

This paper develops an Ito formula for UMD Banach spaces and uses it to establish the existence and regularity of solutions to certain stochastic evolution equations, including the Zakai equation.

Contribution

The authors extend Ito's formula to UMD Banach spaces and apply it to prove solution regularity for stochastic evolution equations like the Zakai equation.

Findings

Established Ito's formula in UMD Banach spaces.

Proved existence of strong solutions for stochastic evolution equations.

Demonstrated regularity of Zakai equation solutions.

Abstract

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.