Adam Andersson

Weak convergence for a spatial approximation of the nonlinear stochastic heat equation

We find the weak rate of convergence of approximate solutions of the nonlinear stochastic heat equation, when discretized in space by a standard finite element method. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation only has a solution in one space dimension. In order to get results for higher dimensions, colored noise is considered here, besides the white noise case where considerably weaker assumptions on the noise term is needed. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.

Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE

We introduce a new family of refined Sobolev–Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.

Weak error analysis for semilinear stochastic Volterra equations with additive noise

We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.

Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be

Duality in refined Watanabe-Sobolev spaces and weak approximations of SPDE

\frac{1}{2}

On the differentiability of solutions of stochastic evolution equations with respect to their initial values

12 for the two-step BDF-Maruyama scheme and for the backward Euler–Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the

Malliavin regularity and weak approximation of semilinear SPDE with L\'evy noise

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