Optimal approximation of stochastic integrals in analytic noise model
This work provides theoretical error bounds for stochastic integration under low-precision computations, relevant for practitioners using GPUs and CPUs for speedup.
The paper studies optimal approximation of stochastic Itô integrals under a noisy information model, showing that the Riemann-Maruyama quadrature achieves error proportional to n^{-ϱ}+δ₁+δ₂, and proving a matching lower bound of C(n^{-ϱ}+δ₁). Numerical experiments on CPU and GPU confirm the theoretical results.
We study approximate stochastic Itô integration of processes belonging to a class of progressively measurable stochastic processes that are Hölder 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 $n^{-\varrho}+δ_1+δ_2$, where $n$ is a number of noisy evaluations of $X$ and $W$, $\varrho\in (0,1]$ is a Hölder exponent of $X$, and $δ_1,δ_2\geq 0$ 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 $C(n^{-\varrho}+δ_1)$. 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.