Sonja Cox

Sonja Cox, Erika Hausenblas

We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence we consider is strong and almost sure uniform convergence, i.e., pathwise convergence. The proofs are based on a previously published perturbation result for such equations.

Sonja Cox, Matas Urbonas

We establish upper bounds for the weak and strong error resulting from a perturbation of the noise driving the stochastic Burgers equation, where we assume the noise to be additive and of trace class and the initial value to be sufficiently regular. More specifically, replacing the covariance operator of the driving noise $Q_1 \in \mathcal{L}_1(L^2)$ in the Burgers equation by a covariance operator $Q_2 \in \mathcal{L}_1(L^2)$ results in a weak error of $\mathcal{O}\big(\| (-A)^{-1^{-} } (Q_1-Q_2) \|_{\mathcal{L}_1(L^2)}\big)$ and a strong error of $\mathcal{O}\big(\big\| (-A)^{-1/2^{-}}\big|Q_1^{1/2} -Q_2^{1/2}\big| \big\|_{\mathcal{L}_2(L^2)}\big)$. Here $\|\cdot \|_{\mathcal{L}_1}$ is the trace class norm, $\|\cdot \|_{\mathcal{L}_2}$ is the Hilbert-Schmidt norm, and $A$ is the one-dimensional Dirichlet Laplacian that represents the leading term in the Burgers equation. In particular, our results provide upper bounds for the weak and strong error arising when approximating the trace class noise by finite-dimensional noise; the rates we obtain reflect the general philosophy that the weak convergence rate should be twice the strong rate.

Sonja Cox, Martin Hutzenthaler, Arnulf Jentzen et al.

We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process which is strongly Hölder continuous in time, then this sequence converges in the strong sense even with respect to much stronger Hölder norms and the convergence rate is essentially reduced by the Hölder exponent. Our first application hereof establishes pathwise convergence rates for spectral Galerkin approximations of stochastic partial differential equations. Our second application derives strong convergence rates of multilevel Monte Carlo approximations of expectations of Banach space valued stochastic processes.

Sonja Cox, Jan van Neerven

We study the splitting scheme associated with the linear stochastic Cauchy problem dU(t) = AU(t) dt + dW(t), where A is the generator of an analytic C_0-semigroup S={S(t)} on a Banach space E and W={W(t)} is a Brownian motion with values in a fractional domain space E_\b associated with A. We prove that if \a,\b,\g,þ\ge 0 are such that \g + þ< 1 and max[0,(\a-\b+þ)] + \g < 1/2, then the approximate solutions U_n (where n is the number of time steps) converge to the solution U in the Holder space C^\g([0,T];E_\a), both in L^p-means and almost surely, with rate 1/n^þ.