TY - JOUR

T1 - Iterated stochastic integrals in infinite dimensions

T2 - approximation and error estimates

AU - Leonhard, Claudine

AU - Rößler, Andreas

PY - 2019/6/15

Y1 - 2019/6/15

N2 - 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, Platen, and Wright (1992) and by Wiktorsson (2001) 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.

AB - 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, Platen, and Wright (1992) and by Wiktorsson (2001) 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.

UR - http://www.scopus.com/inward/record.url?scp=85065710647&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/iterated-stochastic-integrals-infinite-dimensions-approximation-error-estimates

U2 - 10.1007/s40072-018-0126-9

DO - 10.1007/s40072-018-0126-9

M3 - Journal articles

AN - SCOPUS:85065710647

SN - 2194-0401

VL - 7

SP - 209

EP - 239

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

IS - 2

ER -