Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingale noises

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 adapted 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. Finally, simulations complete the paper.