Random Generation of Stochastic Area Integrals

Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

“…For example, the Crank-Nicolson implicit scheme gives an error of order n −2 with h comparable with n −1 . For the equation with noise the results of [3] show that (2) gives an approximation with error of order n −1/2 , provided n 2 h ≤ b < 1 2 ; we shall show that this order of approximation is best possible for schemes of this nature. The main reason why higher order methods do not give improvements is the lack of smoothness of the solution of the equation (1).…”

“…The results just described indicate that solutions of equation (1) can be approximated using scheme (2) to within an average error of order using computational effort of order −6 (since we use O(n 3 ) rectangles and the error is O(n −1/2 )). In the following sections we investigate the extent to which this performance can be improved.…”

“…is measurable with respect to the σ-algebra generated by {W (y, s) : 0 ≤ s < t}, and satisfying sup S Ev(x, t) 2 s)dW (y, s). Straightforward estimates show that T is a contraction with respect to the norm given by v 2 = sup S e −Mt Ev(x, t) 2 .…”

“…A proof of convergence for a simple approximation scheme (see (2) below), and an estimate of the order of convergence, was given by Gyöngy [3]; his results in fact apply to the more general equationu = ∂ 2 u ∂x 2 + f (x, t, u) + σ(x, t, u)Ẇ (x, t). The purpose of the present paper is to investigate to what extent these results are best possible.…”

“…The simplest such approximation is the explicit scheme u j,m+1 = u j,m + n 2 h(u j+1,m + u j−1,m − 2u j,m ) + nσ(u j,m )W j,m (2) for j = 0, 1, . .…”