interlaced euler scheme for stiff systems of stochastic differential equations
In deterministic as well as stochastic models, stiff systems, i.e., systems with vastly different time scales where the fast scales are stable, are very common. It is well known that the implicit Euler method is well suited for stiff deterministic equations (modeled by ODEs) while the explicit Euler is not. In particular, once the fast transients are over, the implicit Euler allows for the choice of time steps comparable to the slowest time scales of the system. In stochastic systems (modeled by SDEs) the picture is more complex. While the implicit Euler has better stability properties over the explicit Euler, it underestimates the stationary variance. In general, one may not expect any method to work successfully by taking time steps of the order of the slowest time scale. We explore the idea of interlacing large implicit Euler steps with a sequence of small explicit Euler steps. In particular, we present our study of a linear test system of SDEs and demonstrate that such interlacing could effectively deal with stiffness. We also discuss the uniform convergence of mean and variance.
stiffness stochastic differential equations implicit euler method uniform convergence explicit euler method