Large deviations for neutral stochastic functional differential equations

Yongqiang Suo, ,Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK, Chenggui Yuan
2020 Communications on Pure and Applied Analysis  
In this paper, under a one-sided Lipschitz condition on the drift coefficient we adopt (via contraction principle) an exponential approximation argument to investigate large deviations for neutral stochastic functional differential equations. Introduction. As is well known, large deviation principle (LDP for short) is a branch of probability theory that deals with the asymptotic behaviour of rare events, and it has a wide range of applications, such as in mathematical finance, statistic
more » ... s, biology and so on. So the LDP for SDEs has been investigated extensively; see, e.g., [1, 2, 16 ] and references therein. From the literature, we know there are two main methods to investigate the LDPs, one method is based on contraction principle in LDPs, that is, it relies on approximation arguments and exponential-type probability estimates; see e.g., [3, 9, 10, 11, 12, 13, 16, 17] and references therein. [9, 13, 17, 19 ] concerned about the LDP for SDEs driven by Brownian motion or Poisson measure, [10] investigated the LDP for invariant distributions of memory gradient diffusions. [11] investigated how rapid-switching behaviour of solution X t affects the small-noise asymptotics of X t -modulated diffusion processes on the certain interval. The other one is weak convergence method, which has also been applied in establishing LDPs for a various stochastic dynamic systems; see e.g., [1, 2, 4, 5, 6, 7] . According to the compactness argument in this method of the solution space of corresponding skeleton equation, the weak convergence is done for Borel measurable functions whose existence is based on Yamada-Watanabe theorem. In [4, 5, 7] , the authors investigated an LDP for SDEs/SPDEs. Compared with the weak convergence method, there are few literature about the LDP for SFDEs, [16] gave result about LDP for SDEs with point delay, and large deviations for perturbed reflected diffusion processes was investigated in [3] . The aim of this paper is to study the LDP for neutral stochastic functional differential equations (NSFDEs), which extends the result in [16] . The structure of this paper is as follows. In section 2, we introduce some preliminary results and notation. In section 3, we state the main results about LDP for NSFDEs and give the corresponding proofs.
doi:10.3934/cpaa.2020103 fatcat:4wrdokwbkjbmdl4vedrybb7xmy