An Adaptive Random Bit Multilevel Algorithm for SDEs

TL;DR

This paper introduces an adaptive multilevel Monte Carlo algorithm using random bits for approximating expectations of SDE solutions, leveraging the Euler scheme and optimal normal distribution approximations.

Contribution

It presents a novel adaptive random bit multilevel algorithm for SDEs that replaces random numbers with random bits, improving efficiency and applicability.

Findings

Comparable accuracy to classical methods in numerical experiments

Efficient random bit approximations of the normal distribution

Potential for reduced randomness requirements in Monte Carlo simulations

Abstract

We study the approximation of expectations $\operatorname{E}(f(X))$ for solutions $X$ of stochastic differential equations and functionals $f$ on the path space by means of Monte Carlo algorithms that only use random bits instead of random numbers. We construct an adaptive random bit multilevel algorithm, which is based on the Euler scheme, the L\'evy-Ciesielski representation of the Brownian motion, and asymptotically optimal random bit approximations of the standard normal distribution. We numerically compare this algorithm with the adaptive classical multilevel Euler algorithm for a geometric Brownian motion, an Ornstein-Uhlenbeck process, and a Cox-Ingersoll-Ross process.