Lp and almost sure convergence of a Milstein scheme for stochastic partial differential equations

In this paper, Lp convergence and almost sure convergence of the Milstein approximation of a partial differential equation of advection–diffusion type driven by a multiplicative continuous martingale is proven. The (semidiscrete) approximation in space is a projection onto a finite dimensional function space. The considered space approximation has to have an order of convergence fitting to the order of convergence of the Milstein approximation and the regularity of the solution. The approximation of the driving noise process is realized by the truncation of the Karhunen–Loève expansion of the driving noise according to the overall order of convergence. Convergence results in Lp and almost sure convergence bounds for the semidiscrete approximation as well as for the fully discrete approximation are provided.