Large Deviation Principle of Numerical Solution

Large deviations describe the asymptotic behavior of small probabilities of rare events on an exponential scale. In Theorem 1.20 , we have established the Freidlin–Wentzell type large deviation principle (LDP) for the exact solution of the SFDE with small noise. In large deviation theory, the decay rates of probabilities associated with rare events are characterized by the large deviation rate function, which is typically defined through a constrained minimization problem and, in general, cannot be expressed explicitly. Therefore, it is of great interest to investigate whether numerical methods can asymptotically preserve the LDPs of the underlying systems, particularly when it comes to predicting rare-event probabilities that are of practical significance. In this chapter, based on the weak convergence approach, we study Freidlin–Wentzell type LDPs for \(\theta \) -EM methods applied to SFDEs with small noise in the infinite time horizon.