Exact Simulation of the Extrema of Stable Processes

TL;DR

This paper introduces an exact simulation algorithm for the maximum of stable processes over a finite interval, utilizing a novel perpetuity equation and dominated coupling from the past, with proven finite coalescence time.

Contribution

It presents a new exact simulation method for stable process extrema using a novel perpetuity equation and DCFTP, with proven finite stopping time.

Findings

Algorithm successfully simulates process maxima accurately.

Finite expected coalescence time is established.

Numerical performance analysis confirms efficiency.

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 L\'evy 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).