A derivative-free Milstein type approximation method for SPDEs covering the non-commutative noise case

Abstract

We propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations (SPDEs) that do not need to fulfill a commutativity condition for the noise term. The newly developed derivative-free Milstein type scheme differs significantly from schemes that are appropriate for the case of commutative noise. As a key result, the new derivative-free Milstein type scheme needs only two stages that are specifically tailored based on a technique that, compared to the original Milstein scheme, allows for a reduction of the computational complexity by one order of magnitude. Moreover, the proposed derivative-free Milstein scheme can flexibly be combined with some approximation method for the involved iterated stochastic integrals. As the main result, we prove the strong L^2-convergence of the introduced derivative-free Milstein type scheme, especially if it is combined with any suitable approximation algorithm for the necessary iterated stochastic integrals. We carry out a rigorous analysis of the error versus computational cost and derive the effective order of convergence for the derivative-free Milstein type scheme in the case that the truncated Fourier series algorithm for the approximation of the iterated stochastic integrals is applied. As a further novelty, we show that the use of approximations of iterated stochastic integrals based on truncated Fourier series together with the proposed derivative-free Milstein type scheme improves the effective order of convergence compared to that of the Euler scheme and the original Milstein scheme. This result is contrary to well known results in the finite dimensional SDE case where the use of merely truncated Fourier series does not improve the effective order of convergence in the L^2-sense compared to that of the Euler scheme.