Three Improvements to Multi-Level Monte Carlo Simulation of SDE Systems

TL;DR

This paper proposes three enhancements to multilevel Monte Carlo methods for solving SDE systems, focusing on cost reduction, alternative discretization, and generalization of antithetic techniques, supported by numerical comparisons.

Contribution

It introduces Ito linearization, an antithetic method without Levy area simulation, and a generalized antithetic approach for MLMC, advancing efficiency and flexibility.

Findings

Cost reduction with Ito linearization for twice differentiable payoffs

Optimal cost-to-error scaling without Levy area simulation

Effective generalization of antithetic methods for various refinement factors

Abstract

We introduce three related but distinct improvements to multilevel Monte Carlo (MLMC) methods for the solution of systems of stochastic differential equations (SDEs). Firstly, we show that when the payoff function is twice continuously differentiable, the computational cost of the scheme can be dramatically reduced using a technique we call `Ito linearization'. Secondly, by again using Ito's lemma, we introduce an alternative to the antithetic method of Giles et. al [M.B. Giles, L. Szpruch. arXiv preprint arXiv:1202.6283, 2012] that uses an approximate version of the Milstein discretization requiring no Levy area simulation to obtain the theoretically optimal cost-to-error scaling. Thirdly, we generalize the antithetic method of Giles to arbitrary refinement factors. We present numerical results and compare the relative strengths of various MLMC-type methods, including each of those presented here.