Positivity of mild solution to stochastic evolution equations with an application to forward rates

TL;DR

This paper establishes a maximum principle for mild solutions to stochastic evolution equations with Lipschitz coefficients and Wiener noise, and applies it to ensure positivity of forward rates in the Heath-Jarrow-Morton model.

Contribution

It introduces a maximum principle for stochastic evolution equations and applies it to guarantee positivity of forward rates in a financial model.

Findings

Maximum principle for stochastic evolution equations proved.

Sufficient conditions for positivity of forward rates established.

Application to Heath-Jarrow-Morton model demonstrated.

Abstract

We prove a maximum principle for mild solutions to stochastic evolution equations with (locally) Lipschitz coefficients and Wiener noise on weighted $L^2$ spaces. As an application, we provide sufficient conditions for the positivity of forward rates in the Heath-Jarrow-Morton model, considering the associated Musiela SPDE on a homogeneous weighted Sobolev space.