Optimal approximation of stochastic integrals in analytic noise model

Abstract

We study approximate stochastic It\^o integration of processes belonging to a class of progressively measurable stochastic processes that are H\"older continuous in the $r$th mean. Inspired by increasingly popularity of computations with low precision (used on Graphics Processing Units -- GPUs and standard Computer Processing Units -- CPU for significant speedup), 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 the error of the Riemann-Maruyama quadrature are proportional to $n^{-\varrho}+\delta_1+\delta_2$, where $n$ is a number of noisy evaluations of $X$ and $W$, $\varrho\in (0,1]$ is a H\"older exponent of $X$, and $\delta_1,\delta_2\geq 0$ 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 $C(n^{-\varrho}+\delta_1)$. Finally, we report numerical experiments performed on both CPU and GPU, that confirm our theoretical findings, together with some computational performance comparison between those two architectures.