Random walk numerical scheme for the steady-state of stochastic differential equations

Abstract

The continuous-time random walk (CTRW) scheme is a time-continuous and space-discretization method to obtain the numerical solution of stochastic differential equations (SDEs). Compared with the traditional time-discretization scheme, it has the advantages of numerical stability and can alleviate the curse of dimensionality. This paper proposes an improved version of the CTRW scheme for the numerical solution of SDEs. By compensating the artificial diffusion caused by the Poisson approximation of the drift term of the SDE, the improved CTRW scheme has significantly better performance in the weak noise case, especially in approximating the invariant probability measure. Numerical studies show that the improved CTRW scheme has more accuracy than the existing one but takes less computation time. In addition, it has better accuracy of the mean holding time. We also modify the hybrid Fokker–Planck solver proposed for the CTRW scheme to compute the invariant probability measure.