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

Abstract

Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisaiton to construct ε-strong simulation (εSS) 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 εSS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.

Discretization of the Lamperti representation of a positive self-similar Markov process

“…One of the reasons is that for an oscillating Lévy process the first passage time over a fixed level has infinite expectation. In this regard we note that [14] recently provided an ε-strong simulation algorithm for the convex minorants of stable meanders, which are closely related to conditioned processes.…”

Unbiased Simulation of Rare Events in Continuous Time

“…[12] contains a related construction for fractional Brownian motion of any Hurst index, and for solutions to SDEs driven by fractional Brownian motion with Hurst index H > 1 2 . An ε-strong algorithm for a process which is not the solution to an SDE is described in [8]. Recent applications of this methodology include [32].…”