Toward Spectral Deferred Correction Methods for Stochastic Differential Equations
The numerical solution of stochastic differential equations is computationally very demanding. Often it is of interest to evaluate certain moments, e.g., the expectation, by Monte Carlo sampling, thus requiring the computation of numerous realizations. The discretization error of the numerical scheme induces bias in the computed Monte Carlo estimator. As popular numerical schemes, like Euler-Maruyama or Milstein, are low-order, even reasonably coarse error bounds for the simulation are costly to achieve. While the evaluation of moments requires only weak convergence, in other applications like parameter estimation, strong convergence is important, but higher-order is typically harder to obtain in the strong than in the weak sense. In this paper we develop an efficient method with the potential of achieving higher strong and weak convergence order. It is based on the smooth approximation of Brownian bridges combined with spectral deferred correction methods well known for the numerical solution of ordinary differential equations. We discuss convergence, and present numerical results.