A Large Deviation Principle for Martingales over Brownian Filtration

TL;DR

This paper proves a large deviation principle for scaled martingales over Brownian filtration, linking the rate function to Wiener-Itô chaos and employing advanced stochastic analysis tools.

Contribution

It introduces a large deviation principle for martingales over Brownian filtration, with a novel rate function expressed via Wiener-Itô chaos decomposition.

Findings

Established a large deviation principle for scaled martingales.

Identified the rate function using Wiener-Itô chaos decomposition.

Developed a continuity theorem for large deviations in this context.

Abstract

In this article we establish a large deviation principle for the family {\nu_{\epsilon}:\epsilon \in (0,1)} of distributions of the scaled stochastic processes {P_{-\log\sqrt{\epsilon}}Z_t}_{t\leq 1}, where (Z_t)_{t\in \lbrack 0,1]} is a square-integrable martingale over Brownian filtration and (P_t)_{t\geq 0} is the Ornstein-Uhlenbeck semigroup. The rate function is identified as well in terms of the Wiener-It\^{o} chaos decomposition of the terminal value Z_{1}. The result is established by developing a continuity theorem for large deviations, together with two essential tools, the hypercontractivity of the Ornstein-Uhlenbeck semigroup and Lyons' continuity theorem for solutions of Stratonovich type stochastic differential equations.