Geometrically Convergent Simulation of the Extrema of Lévy Processes

Jorge Ignacio González Cázares (), Aleksandar Mijatović () and Gerónimo Uribe Bravo ()
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Jorge Ignacio González Cázares: The Alan Turing Institute, NW1 2DB London, United Kingdom; Department of Statistics, University of Warwick, CV4 7AL Coventry, United Kingdom
Aleksandar Mijatović: The Alan Turing Institute, NW1 2DB London, United Kingdom; Department of Statistics, University of Warwick, CV4 7AL Coventry, United Kingdom
Gerónimo Uribe Bravo: Instituto de Matemáticas, Universidad Nacional Autónoma de México, 04510 Ciudad de México, México

Mathematics of Operations Research, 2022, vol. 47, issue 2, 1141-1168

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 Lévy 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 ≥ 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 nonasymptotic 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 ϵ − 2 if the mean squared error is at most ϵ 2 ) for locally Lipschitz and barrier-type functions of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.

Keywords: Primary: 60G51; 65C05; supremum; Lévy process; simulation; Monte Carlo estimation; geometric convergence; multilevel Monte Carlo (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:2:p:1141-1168

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