Abstract
Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992. https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001. https://doi.org/10.1214/aoap/1015345301) for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional Q-Wiener process in case of a trace class operator Q given the increments of the Q-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator Q. This difference arises as the stochastic process is of infinite dimension.
Highlights
In order to obtain higher convergence rates for numerical schemes for stochastic differential equations, in general, we need to incorporate the information contained in iterated integrals
We show that the method that is superior in the setting of an infinite dimensional Q-Wiener process cannot be uniquely determined in general but is dependent on the covariance operator Q
D → ∞ for both algorithms that we proposed in the last sections with differing orders, respectively
Summary
In order to obtain higher convergence rates for numerical schemes for stochastic differential equations, in general, we need to incorporate the information contained in iterated integrals. We want to emphasize that the algorithms developed for the approximation of iterated stochastic integrals in the setting of SDEs are designed for some fixed finite number K of driving Brownian motions and that the approximation error (4) even involves this number K as a constant. We derive two algorithms for the approximation of iterated integrals of type (7) based on the methods developed for the finite dimensional setting by Kloeden, Platen, and Wright [4] and by Wiktorsson [9] as well as on [5] for the infinite dimensional case These algorithms allow for the first time to implement higher order schemes for SPDEs that do not possess commutative noise and include the algorithms that can be used for finite dimensional SDEs as a special case.
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