Numerical methods for Langevin-type SPDE: an implicit Milstein approach and multilevel Monte Carlo techniques

Abstract

In this work, we investigate the numerical approximation of degenerate Langevin-type stochastic partial differential equations (SPDEs) in two spatial dimensions. These SPDEs arise in stochastic dynamics and mathematical finance, among other applications. In order to handle the mixed deterministic-stochastic structure of the equation and the degeneracy of the differential operator, we propose a semi-implicit Milstein finite difference scheme for the numerical solution. Through the Fourier analysis of the mean-square stability and convergence, we derive explicit conditions on the coefficients under which the scheme is stable, jointly with explicit convergence rates in terms of the discretization parameters. We further embed the proposed scheme within a Multilevel Monte Carlo (MLMC) framework to reduce the computational cost associated with SPDE simulations, and we derive its theoretical computational complexity. Numerical experiments confirm theoretical convergence rates and show that the MLMC strategy achieves an accuracy comparable to standard Monte Carlo at a fraction of the computational cost, reducing the complexity from $\mathcal{O}(\varepsilon^{-5})$ to $\mathcal{O}(\varepsilon^{-3})$ for a target root-mean-square error $\varepsilon$. These results show that combining semi-implicit Milstein schemes with MLMC techniques provides an effective approach for the numerical simulation of Langevin-type SPDEs.