In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler–Maruyama approximation. Finally, simulations complete the paper.
%0 Journal Article
%1 barth2012milstein
%A Barth, Andrea
%A Lang, Annika
%D 2012
%J Applied Mathematics & Optimization
%K 2012 B07 sfbtrr161
%N 3
%P 387-413
%R 10.1007/s00245-012-9176-y
%T Milstein Approximation for Advection-diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises
%U http://dx.doi.org/10.1007/s00245-012-9176-y
%V 66
%X In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler–Maruyama approximation. Finally, simulations complete the paper.
@article{barth2012milstein,
abstract = {In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler–Maruyama approximation. Finally, simulations complete the paper.},
added-at = {2020-03-05T13:51:18.000+0100},
author = {Barth, Andrea and Lang, Annika},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/27692614e7f3fb3ff47f437dd43c10f84/leonkokkoliadis},
coden = {AMOMBN},
doi = {10.1007/s00245-012-9176-y},
fjournal = {Applied Mathematics and Optimization. An International Journal with
Applications to Stochastics},
interhash = {905020a555ec0b881967d6b0ce9b6ca1},
intrahash = {7692614e7f3fb3ff47f437dd43c10f84},
issn = {0095-4616},
journal = {Applied Mathematics & Optimization},
keywords = {2012 B07 sfbtrr161},
mrclass = {65C30 (35R60 60H15 60H35 65M75)},
mrnumber = {2996432},
number = 3,
owner = {barthaa},
pages = {387-413},
timestamp = {2020-03-05T12:51:18.000+0100},
title = {Milstein Approximation for Advection-diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises},
url = {http://dx.doi.org/10.1007/s00245-012-9176-y},
volume = 66,
year = 2012
}
