Numerical approximations of one-point large deviations rate functions of stochastic differential equations with small noise

In this paper, we study the numerical approximation of the one-point large deviations rate functions of nonlinear stochastic differential equations (SDEs) with small noise. We show that the stochastic $\theta$-method satisfies the one-point large deviations principle with a discrete rate function for sufficiently small step-size, and present a uniform error estimate between the discrete rate function and the continuous one on bounded sets in terms of step-size. It is proved that the convergence orders in the cases of multiplicative noises and additive noises are $1/2$ and $1$ respectively. Based on the above results, we obtain an effective approach to numerically approximating the large deviations rate functions of nonlinear SDEs with small time. To the best of our knowledge, this is the first result on the convergence rate of discrete rate functions for approximating the one-point large deviations rate functions associated with nonlinear SDEs with small noise.