# Optimal approximation of stochastic integrals in analytic noise model

## TL;DR

This paper investigates the optimal approximation of stochastic Itô integrals under an analytic noise model, establishing error bounds for noisy evaluations and demonstrating the impact of low-precision computations on accuracy and performance.

## Contribution

It introduces a new analytic noise model for stochastic integration and derives error bounds for approximation algorithms considering low-precision evaluations.

## Key findings

- Error bounds are proportional to $n^{-
ho} + ext{precision parameters}$.
- Any algorithm with at most $n$ evaluations has a lower bound error of $C(n^{-
ho} + ext{precision})$.
- Numerical experiments confirm theoretical error bounds and compare CPU and GPU performance.

## 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.