Singular perturbations and asymptotic expansions for SPDEs with an application to term structure models

TL;DR

This paper investigates how solutions to certain linear stochastic evolution equations depend on a small parameter, deriving asymptotic expansions and applying these results to a financial model of forward rates.

Contribution

It provides new asymptotic expansion techniques for SPDE solutions with a small perturbation parameter, with applications to financial mathematics.

Findings

Derived asymptotic expansions for SPDE solutions as perturbation parameter approaches zero

Established convergence and error bounds for the expansions

Applied results to a financial model of forward rate dynamics

Abstract

We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type $A+\varepsilon G$, on the parameter $\varepsilon$. In particular, we study the limit and the asymptotic expansions in powers of $\varepsilon$ of these solutions, as well as of functionals thereof, as $\varepsilon \to 0$, with good control on the remainder. These convergence and series expansion results are then applied to a parabolic perturbation of the Musiela SPDE of mathematical finance modeling the dynamics of forward rates.