On large deviations for small noise It\^o processes

TL;DR

This paper establishes a large deviation principle for small noise Itô processes, including degenerate cases and systems with memory, using the Dupuis-Ellis weak convergence approach, with applications to positive diffusions.

Contribution

It introduces a novel large deviation framework for degenerate Itô SDEs with parameter-dependent coefficients, extending existing results to more general systems.

Findings

Large deviation principle derived for degenerate Itô processes.

Applicable to systems with memory and positive diffusions.

Uses the Dupuis-Ellis weak convergence method.

Abstract

The large deviation principle in the small noise limit is derived for solutions of possibly degenerate It\^o stochastic differential equations with predictable coefficients, which may depend also on the large deviation parameter. The result is established under mild assumptions using the Dupuis-Ellis weak convergence approach. Applications to certain systems with memory and to positive diffusions with square-root-like dispersion coefficient are included.