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 ε → 0. In other words, we consider P(‖X ^ε − ϕ‖ < δ, ‖u ^ε− ψ‖ < τ) where u ^ε (t) and ψ (t) are the occupation times of X ^ε and ϕ in the positive half space {x ∈ ℝ^d : x ₁ > 0} respectively. As a consequence, a unified approach of the lower-level large deviation principle for the law of X ^ε (·) P(‖X ^ε − ϕ‖ < τ) can be obtained.