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In this short note, a direct proof of L2 convergence of an Euler–Maruyama approximation of a Zakai equation driven by a square integrable martingale is shown. The order of convergence is as known for real-valued stochastic differential equations and for less general driving noises O(Δt) for a time discretization step size Δt

Mean square convergence of a semidiscrete scheme for SPDEs of Zakai type driven by square integrable martingales