Large and Moderate Deviations for Finite Dimensional Systems

Abstract

In this chapter we use the abstract sufficient conditions from Chap. 9 to prove large and moderate deviation principles for small noise finite dimensional jump-diffusions. We will consider only Laplace principles rather than uniform Laplace principles, since, as was noted in Chap. 9, the extension from the nonuniform to the uniform case is straightforward. The first general results on large deviation principles for jump-diffusions of the form considered in this chapter are due to Wentzell [245–248] and Freidlin and Wentzell [140]. The conditions for an LDP identified in the current chapter relax some of the assumptions made in these works. Results on moderate deviation principles in this chapter are based on the recent work [41]. We do not aim for maximal generality, and from the proofs it is clear that many other models (e.g., time inhomogeneous jump diffusions, SDEs with delay) can be treated in an analogous fashion.