Geometrically Convergent Simulation of the Extrema of L\'{e}vy Processes

TL;DR

This paper introduces a new simulation algorithm for Lévy processes that achieves geometric error decay, enabling efficient and accurate estimation of extrema-related functionals with proven theoretical guarantees.

Contribution

The authors develop a novel simulation method with geometric error decay for Lévy process extrema, including error analysis, CLT, confidence intervals, and optimal multilevel Monte Carlo complexity.

Findings

Error decays geometrically with computational cost

Multilevel Monte Carlo achieves optimal complexity of order ε^{-2}

Algorithm performs well in numerical experiments

Abstract

We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general L\'evy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in $L^p$ (for any $p\geq 1$) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct non-asymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We prove that the multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $\epsilon^{-2}$ if the mean squared error is at most $\epsilon^2$) for locally Lipschitz and barrier-type functionals of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.