An Adaptive Random Bit Multilevel Algorithm for SDEs
This work addresses the problem of reducing reliance on high-quality random numbers in Monte Carlo simulations for SDEs, which is important for applications where random bits are cheaper or more available.
The paper develops an adaptive random bit multilevel algorithm for approximating expectations of SDE solutions, using only random bits instead of random numbers. Numerical comparisons show that the algorithm achieves similar accuracy to the classical multilevel Euler method while using fewer random bits.
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 Lévy-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.