Almost sure convergence of a Galerkin approximation for SPDEs of Zakai type driven by square integrable martingales

Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises

“…Here, we combine and extend results from [6,33,35,36], and [34] and derive L 2 and almost sure convergence for an approximation scheme with a not necessarily equidistant time discretization. The increased convergence of order one in the time discretization is derived by adding an extra term to the well-known Euler-Maruyama scheme, which itself just leads to convergence of order O(k 1/2 ).…”

“…Furthermore, in [6] a (semidiscrete) space approximation and a fully discrete approximation using a Galerkin method in space and a backward Euler approach in time were introduced. A space approximation for an equation driven by a-not necessarily continuous-square integrable martingale was done in [34].…”

Finite difference schemes for linear stochastic integro-differential equations

“…20) and(4.22), we have(4.19) for m = 0. The case m > 0 follows from the case m = 0, since for a multi-index γ, we have∂ γ (Iu − I h u) = I∂ γ u − I h ∂ γ u.…”

“…In [16], E. Hausenblas considers finite element approximations of linear SPDEs in polyhedral domains D driven by Poisson noise of impulsive-type and derives L p (Ω) error estimates in the L p (D)-norm. In a more recent work [20], A. Lang studied semi-discrete Galerkin approximation schemes for SPDEs of advection diffusion type in bounded domains D driven by cádlág square integrable martingales in a Hilbert space. A. Lang showed that the rate of convergence in the L p (Ω) and almost-sure sense in the L 2 (D)-norm is of order two for a finite-element Galerkin scheme.…”

“…The articles [20], [1], [17], and [16] do not address the approximation of equations with non-local operators in the drift and noise. There is, however, some work in the literature on deterministic non-local differential equations.…”