Numerical approximation for stochastic differential equations with state-dependent fast switching
This paper aims to develop efficient numerical approximations for a class of stochastic differential equations with state-dependent fast switching processes. The direct Euler--Maruyama (EM) scheme fails when the scaling parameter is small. Based on the heterogeneous multiscale method of \cite{e2005analysis}, we propose three algorithms and prove their strong $L^p$-convergence with explicit rates for any $p\geq 2$. In the first algorithm, we combine the averaging principle with an EM scheme for the averaged equation, where the invariant measure of the Markov chain can be explicitly obtained by solving a linear system. However, computing this invariant measure incurs cubic cost as the number of switching states increases. To avoid solving large linear systems, we approximate the invariant measure instead. Thus the second and third algorithms both combine a macroscopic EM scheme for a modified averaged equation with micro-solvers that estimate the averaged drift. More precisely, in the second algorithm, a discrete-time Markov chain is simulated with a micro time step, and the averaged drift is obtained by averaging over finitely many micro transitions. However, both the first and second algorithms only work when the switching process has finite states. Therefore, we introduce a third algorithm that allows for switching processes with countably infinite states, in which exact continuous-time Markov chain is generated via the Gillespie algorithm, and the averaged drift is computed by exact time averaging over a specified interval. Numerical experiments verify the theoretical results and demonstrate the computational advantages of these three algorithms.