Large deviations for invariant measures of stochastic differential equations with jumps

ABSTRACT

Large deviation principle (LDP) for the invariant measures of stochastic differential equations with jumps is obtained. First, we prove $\left\{X^{\epsilon }\right(t){\}}_{\epsilon >0}$ given as the solutions of the stochastic differential equations have invariant measures $\left\{{\mu }^{\epsilon }\right(\cdot ){\}}_{\epsilon >0}$. In order to ensure the uniqueness of the invariant measures, the strong Feller property and irreducibility for the Markov semigroup are proved. Moreover, the LDP for the invariant measures $\left\{{\mu }^{\epsilon }\right(\cdot ){\}}_{\epsilon >0}$ is established. The proof of the LDP for ${\mu }^{\epsilon }$ is based on the LDP for $\left\{X^{\epsilon }\right(t){\}}_{\epsilon >0}$, which have been studied by Budhiraja et al. [5].

KEYWORDS:

• Large deviation
• invariant measure
• strong Feller property
• irreducibility
• Poisson random measure

2010 MR SUBJECT CLASSIFICATIONS:

• 60F60
• 60H10
• 60A10

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research of this author was supported by the National Natural Science Foundation of China (Grant Nos. 11671034, 11771327) and the Beijing Natural Science Foundation (Grant No. 1172001).