Large deviations principle for stochastic delay differential equations with super-linearly growing coefficients

TL;DR

This paper proves a large deviations principle for stochastic delay differential equations with super-linearly growing coefficients, extending previous results to include growth in both delay and state variables using the weak convergence method.

Contribution

It establishes the LDP for SDDEs with super-linear growth in both delay and state variables, broadening the scope of existing theoretical results.

Findings

Established the LDP for SDDEs with super-linear coefficients.

Developed uniform moment estimates for controlled equations.

Extended LDP results to include growth in delay and state variables.

Abstract

We utilize the weak convergence method to establish the Freidlin--Wentzell large deviations principle (LDP) for stochastic delay differential equations (SDDEs) with super-linearly growing coefficients, which covers a large class of cases with non-globally Lipschitz coefficients. The key ingredient in our proof is the uniform moment estimate of the controlled equation, where we handle the super-linear growth of the coefficients by an iterative argument. Our results allow both the drift and diffusion coefficients of the considered equations to super-linearly grow not only with respect to the delay variable but also to the state variable. This work extends the existing results which develop the LDPs for SDDEs with super-linearly growing coefficients only with respect to the delay variable.