Optimal approximation of stochastic integrals in analytic noise model

摘要

We study approximate stochastic Itô integration of processes belonging to a class of progressively measurable stochastic processes that are Hölder continuous in the rth mean.Inspired by increasing popularity of computations with low precision (used on Graphics Processing Units – GPUs and standard Computer Processing Units – CPUs), we introduce a suitable analytic noise model of standard noisy information about X and W. In this model we show that the upper bounds on error of the Riemann–Maruyama quadrature are proportional to n−ϱ+δ1+δ2, where n is a number of noisy evaluations of X and W, ϱ ∈ (0, 1] is a Hölder exponent of X, and δ1, δ2 ≥ 0 are precision parameters for values of X and W, respectively. Moreover, we show that the error of any algorithm based on at most n noisy evaluations of X and W is at least C(n−ϱ+δ1). Finally, we report numerical experiments performed on both CPU and GPU, together with some computational performance comparison between those two architectures.