Exact Simulation of the Extrema of Stable Processes

Abstract

We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Lévy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).

arXiv comment26 pages, 3 figures, Julia implementation of the exact simulation algorithm is in the GitHub repository: https://github.com/jorgeignaciogc/StableSupremum.jl

References in corpus (4)

• The law of the supremum of a stable Lévy process with no negative jumps
• On extrema of stable processes
• The convex minorant of a Lévy process
• Predicting the ultimate supremum of a stable Lévy process with no negative jumps

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• Joint density of a stable process and its supremum: regularity and upper bounds
• Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Lévy Processes with Infinite Activities