Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

Abstract

In this paper, employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.

Moderate Averaged Deviations for a Multi-Scale System with Jumps and Memory

“…. , t ∆ the following: The convergence above follows from the choice of the parametrization ∆ = ∆(ε) fixed in (52) and α constructed in Proposition 3. in (44).…”

“…The proof is given in Section A.2.1 of Appendix A. Lemma 3. For every ε > 0, let R(ε) be fixed as in Proposition 5 and ∆(ε) given by (52). Then, the following convergence holds,…”

“…We refer the reader to the works [49,50] where the authors apply Freidlin-Wentzell types of LDPs to the study of the first exit time problem in the small noise limit for Gaussian diffusions with delay. For the application of the weak convergence approach in the establishment of MDPs to stochastic differential delay equations, we mention the works [51,52].…”

Moderate deviations for rough differential equations

“…The following is a partial list of preceding works on MDPs of this kind. MDPs for various stochastic systems such as jump-type SDEs [2,3], SDEs with delay [21], some financial models [15], stochastic Hamiltonian systems [24], slow-fast systems [9,10,12,16], and Volterra-type SDEs [14,17] have already been proved. For MDPs for stochastic PDEs, see [18,22,23] among others.…”