Pathwise large deviations for white noise chaos expansions

Abstract

We consider a family of continuous processes ${\{X^{\mathit{\epsilon }}\}}_{\mathit{\epsilon }>0}$ which are measurable with respect to a white noise measure, take values in the space of continuous functions $C({[0,1]}^d:\mathbb{R})$, and have the Wiener chaos expansion

$$X^{\mathit{\epsilon }}=\sum _{n=0}^{\mathrm{\infty }}{\mathit{\epsilon }}^n{I}_n\left(f_n^{\mathit{\epsilon }}\right).$$

We provide sufficient conditions for the large deviations principle of ${\{X^{\mathit{\epsilon }}\}}_{\mathit{\epsilon }>0}$ to hold in $C({[0,1]}^d:\mathbb{R})$, thereby refreshing a problem left open by Pérez–Abreu (1993) in the Brownian motion case. The proof is based on the weak convergence approach to large deviations: it involves demonstrating the convergence in distribution of certain perturbations of the original process, and thus the main difficulties lie in analysing and controlling the perturbed multiple stochastic integrals. Moreover, adopting this representation offers a new perspective on pathwise large deviations and induces a variety of applications thereof.