A Large Deviation Principle for Martingales over Brownian Filtration

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.