Large Deviation of Diffusion Processes with Discontinuous Drift

Abstract

For the system of d-dim stochastic differential equations $$ \left\{ {\begin{array}{*{20}{c}} {d{X^ \in }\left( t \right) = b\left( {{X^ \in }\left( t \right)} \right)dt + \in dW\left( t \right),t \in \left[ {0,1} \right]} {{X^ \in }\left( 0 \right) = {x^0} \in {R^d}} \end{array}} \right.$$ where b is smooth except possibly along the hyperplane x 1 = 0, we shall consider the large deviation principle for the law of the solution diffusion process and its occupation time as e → 0. In other words, we consider P(‖X e − ϕ‖ 0} respectively. As a consequence, a unified approach of the lower-level large deviation principle for the law of X e (·) P(‖X e − ϕ‖ < τ) can be obtained.