  The minimum action method (MAM) is an effective approach to numerically
solving minimums and minimizers of Freidlin--Wentzell (F-W) action functionals,
which is used to study the most probable transition path and probability of the
occurrence of transitions for stochastic differential equations (SDEs) with
small noise. In this paper, we focus on MAMs based on a finite difference
method, and present the convergence analysis of minimums and minimizers of the
discrete F-W action functional. The main result shows that the convergence
orders of the minimum of the discrete F-W action functional in the cases of
multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our
main result also reveals the convergence of the stochastic $\theta$-method for
SDEs with small noise in terms of large deviations.
