Random Bit Multilevel Algorithms for Stochastic Differential Equations

TL;DR

This paper develops and analyzes random bit multilevel algorithms for approximating expectations of solutions to stochastic differential equations, demonstrating their near-equivalence in power to traditional randomized methods.

Contribution

It introduces a novel random bit multilevel Euler algorithm and establishes upper and lower bounds, showing its effectiveness compared to general randomized algorithms.

Findings

The proposed algorithm achieves near-optimal error and cost bounds.

Random bit algorithms are nearly as powerful as traditional randomized algorithms.

Matching lower bounds are established up to a logarithmic factor.

Abstract

We study the approximation of expectations $\E(f(X))$ for solutions $X$ of SDEs and functionals $f \colon C([0,1],\R^r) \to \R$ by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals $f$ from the Lipschitz class w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms.