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

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