Large deviation principle for stochastic integrals and stochastic differential equations driven by infinite-dimensional semimartingales

Abstract

The paper concerns itself with establishing large deviation principles for a sequence of stochastic integrals and stochastic differential equations driven by general semimartingales in infinite-dimensional settings. The class of semimartingales considered is broad enough to cover Banach space-valued semimartingales and the martingale random measures. Simple usable expressions for the associated rate functions are given in this abstract setup. As illustrated through several concrete examples, the results presented here provide a new systematic approach to the study of large deviation principles for a sequence of Markov processes.MSC 2010 subject classifications: 60F10, 60G51, 60H05, 60H10, 60J25, 60J60.

Large Deviations for Stochastic Differential Equations Driven by Semimartingales

“…A natural question is: if the family of controls and noise {(X , U )} >0 satisfies the large deviation principle, is it true that the family of solutions {Y } >0 also satisfies the large deviation principle? A similar question has been investigated in [8,7]. The former considered the case that U ≡ 0 (which is not essential), and the latter concern itself with the infinite-dimensional case.…”

“…Besides, we will also surrender the uniform exponential tightness condition. The UET condition given in [8,7] is aiming to establish the analogue results of weak convergence into the large deviations setting, more precisely, the large deviation principle for stochastic integrals with respect to semimartingales. One can say that the UET condition is an exponential version of the uniform tightness (UT) condition in the context of weak convergence, referring to [10, Section VI.6] for the latter.…”

“…We follow the lines of [7,Theorem 6.1]. By the contraction principle, the family (X , U , F (Y )) obeys the LDP with good rate funtion But the contraction principle yields…”

Large Deviations for SDE driven by Heavy-tailed Lévy Processes

“…The relationship between uniformly exponential tightness and the large deviations principle also gives rise to results on SDEs driven by semimartingales. [20,21] However, an exponential integrability condition on the Lévy measure ν is unavoidably in their works, which is…”

Rolling with Random Slipping and Twisting: A Large Deviation Point of View

“…provided that the family of noise {Y } >0 satisfies the large deviation principle. A similar problem of large deviations for SDE (1.3) was investigated in [12,11]. In both papers, the exponentially tight assumption on the family {(X , Y )} was proposed to prove the LDP for the solution family {X }, provided that the LDP holds for {Y }.…”