Large Deviations for a Class of Anticipating Stochastic Differential Equations

Abstract

Consider the family of perturbed stochastic differential equations on Rd, Xεt=Xε0+ε√∫t0σ(Xεs)∘dWs+∫t0b(Xεs)ds, ε>0, defined on the canonical space associated with the standard k-dimensional Wiener process W. We assume that {Xε0,ε>0} is a family of random vectors not necessarily adapted and that the stochastic integral is a generalized Stratonovich integral. In this paper we prove large deviations estimates for the laws of {Xε.,ε>0}, under some hypotheses on the family of initial conditions {Xε0,ε>0}.