Stochastic Evolution Equations in Banach Spaces and Applications to Heath-Jarrow-Morton-Musiela Equation

TL;DR

This paper establishes existence and uniqueness results for stochastic evolution equations in Banach spaces and applies these findings to the Heath-Jarrow-Morton-Musiela equation, including conditions for invariant measures.

Contribution

It develops a framework for solving stochastic evolution equations in Banach spaces and applies it to the HJMM equation, providing new existence, uniqueness, and invariant measure results.

Findings

Proved existence and uniqueness of solutions in Banach spaces.

Applied abstract results to the HJMM equation.

Identified conditions for invariant measures.

Abstract

In this paper we study the stochastic evolution equation (1.1) in martingale-type 2 Banach spaces (with the linear part of the drift being only a generator of a C0-semigroup). We prove the existence and the uniqueness of solutions to this equation. We apply the abstract results to the Heath-Jarrow-Morton-Musiela (HJMM) equation (6.3). In particular, we prove the existence and the uniqueness of solutions to the latter equation in the weighted Lebesgue and Sobolev spaces respectively. We also find a sufficient condition for the existence and the uniqueness of an invariant measure for the Markov semigroup associated to equation (6.3) in the weighted Lebesgue spaces.