An Analysis of the Milstein Scheme for SPDEs without a Commutative Noise Condition

TL;DR

This paper demonstrates that the Milstein scheme can be effectively applied to SPDEs without commutative noise by integrating recent approximation methods for iterated stochastic integrals, analyzing convergence order and computational costs.

Contribution

It extends the Milstein scheme to non-commutative SPDEs using new approximation methods, maintaining convergence order and comparing efficiency with other schemes.

Findings

Milstein scheme maintains convergence order with new integral approximations.

Combination of Milstein with approximation methods can be computationally advantageous.

The scheme outperforms exponential Euler in certain SPDE parameter regimes.

Abstract

In order to approximate solutions of stochastic partial differential equations (SPDEs) that do not possess commutative noise, one has to simulate the involved iterated stochastic integrals. Recently, two approximation methods for iterated stochastic integrals in infinite dimensions were introduced in C. Leonhard and A. R\"o{\ss}ler: Iterated stochastic integrals in infinite dimensions: approximation and error estimates, Stoch. Partial Differ. Equ. Anal. Comput., 7(2): 209-239 (2019). As a result of this, it is now possible to apply the Milstein scheme by Jentzen and R\"ockner: A Milstein scheme for SPDEs, Found. Comput. Math., 15(2): 313-362 (2015) to equations that need not fulfill the commutativity condition. We prove that the order of convergence of the Milstein scheme can be maintained when combined with one of the two approximation methods for iterated stochastic integrals. However, we also have to consider the computational cost and the corresponding effective order of convergence for a meaningful comparison with other schemes. An analysis of the computational cost shows that, in dependence on the equation, a combination of the Milstein scheme with both of the two methods may be the preferred choice. Further, the Milstein scheme is compared to the exponential Euler scheme and we show for different SPDEs depending on the parameters describing, e.g., the regularity of the equation, which one of the schemes achieves the highest effective order of convergence.