On the positivity of local mild solutions to stochastic evolution equations

TL;DR

This paper establishes conditions under which solutions to certain stochastic evolution equations remain positive, with applications to financial models like the Heath-Jarrow-Morton framework.

Contribution

It provides new sufficient conditions for the positivity of mild solutions to stochastic evolution equations on Hilbert spaces, including applications to financial modeling.

Findings

Positivity of solutions is guaranteed under specific coefficient conditions.

Application to the Heath-Jarrow-Morton model confirms positivity of forward rates.

Framework can be used for other stochastic PDEs with positivity constraints.

Abstract

We provide sufficient conditions on the coefficients of a stochastic evolution equation on a Hilbert space of functions driven by a cylindrical Wiener process ensuring that its mild solution is positive if the initial datum is positive. As an application, we discuss the positivity of forward rates in the Heath-Jarrow-Morton model via Musiela's stochastic PDE.