On Some Degenerate Large Deviation Problems

Abstract

This paper concerns the issue of obtaining the large deviation principle for solutions of stochastic equations with possibly degenerate coefficients. Specifically, we explore the potential of the methodology that consists in establishing exponential tightness and identifying the action functional via a maxingale problem. In the author's earlier work it has been demonstrated that certain convergence properties of the predictable characteristics of semimartingales ensure both that exponential tightness holds and that every large deviation accumulation point is a solution to a maxingale problem. The focus here is on the uniqueness for the maxingale problem. It is first shown that under certain continuity hypotheses existence and uniqueness of a solution to a maxingale problem of diffusion type are equivalent to Luzin weak existence and uniqueness, respectively, for the associated idempotent Ito equation. Consequently, if the idempotent equation has a unique Luzin weak solution, then the action functional is specified uniquely, so the large deviation principle follows. Two kinds of application are considered. Firstly, we obtain results on the logarithmic asymptotics of moderate deviations for stochastic equations with possibly degenerate diffusion coefficients which, as compared with earlier results, relax the growth conditions on the coefficients, permit certain non-Lipshitz-continuous coefficients, and allow the coefficients to depend on the entire past of the process and to be discontinuous functions of time. The other application concerns multiple-server queues with impatient customers.