$\varepsilon$-strong simulation of the convex minorants of stable processes and meanders

TL;DR

This paper develops $ ext{ extsterling}$-strong simulation algorithms for the convex minorants of stable processes and meanders, using marked Dirichlet processes to characterize their laws and proving finite exponential moments for the algorithms' running times.

Contribution

It introduces novel $ ext{ extsterling}$-strong simulation methods for stable process convex minorants based on Dirichlet process characterizations, with proven efficiency and implementation.

Findings

Algorithms have finite exponential moments for running times.

Numerical examples confirm convergence and efficiency.

Applicable to a broad class of Lévy processes including stable and symmetric.

Abstract

Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of L\'evy processes, which includes subordinated stable and symmetric L\'evy processes. We apply this characterisaiton to construct $\varepsilon$-strong simulation ($\varepsilon$SS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our $\varepsilon$SS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.