Erratum: Almost sure convergence of a semi-discrete Milstein scheme for SPDEs of Zakai type

Consistency and stability of a Milstein–Galerkin finite element scheme for semilinear SPDE

“…Below we show that the operators R k are well-defined under Assumptions 3.5 and 3.7 for all k ∈ (0, T ]. Further, under these conditions it holds that R k [X k ] = 0 ∈ G p (T k ) for all k ∈ (0, T ], where X k ∈ G p (T k ) is the discrete time stochastic process generated by the numerical scheme (22). The mappings R k are therefore called residual operators associated to the numerical scheme (22).…”

“…, denotes the family of grid functions generated by the numerical scheme (22) and X is the mild solution to (2).…”

“…In Lemma 3.11 it is shown that the local truncation error is dominated by a sum of five terms. The first one is concerned with the distance between the initial conditions of the SPDE (2) and the numerical scheme (22). The next three summands are concerned with the error originating from replacing the analytic semigroup S(t), t ∈ [0, T ], by the family of bounded linear operators S k .…”

“…Consequently, this observation has spurred the study of an infinite dimensional analogue of the Milstein scheme and first results have been achieved for a temporal semidiscretization of linear SPDEs in [21,22]. Afterwards, the Milstein scheme has been combined with Galerkin finite element methods and extended to more general types of driving noises in [1,3], while it was applied to semilinear SPDEs in [13], but only with spectral Galerkin methods.…”

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

“…For Wiener noise, fully discrete approximations of the solution of Eq. (1.1) were already studied in [13], while higher order schemes were presented in [35,36] for a time approximation. 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.…”