A Large Deviation Principle for Martingales over Brownian Filtration

Details A Large Deviation Principle for Martingales over Brownian Filtration A Large Deviation Principle for Martingales over Brownian Filtration

2009-04-03

arxiv.org/abs/0904.0547v1

Mathematics

Probability

47 pages

Scientific paper

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.

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