Optimal Pointwise Approximation of a Linear Stochastic Heat Equation with Additive Space-Time White Noise

Abstract

We consider a linear stochastic heat equation on the spatial domain ]0, 1[ with additive space-time white noise, and we study approximation of the mild solution at a fixed time instance. We show that a drift-implicit Euler scheme with a non-equidistant time discretization achieves the order of convergence N -1/2, where N is the total number of evaluations of one-dimensional components of the driving Wiener process. This order is best possible and cannot be achieved with an equidistant time discretization.