Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function

TL;DR

This paper introduces a fast, exact simulation algorithm for the first passage time of a tempered stable subordinator over a non-increasing function, with proven efficiency and practical numerical demonstrations.

Contribution

The authors develop a novel algorithm that efficiently simulates first passage times of tempered stable subordinators with explicit complexity bounds and practical implementation.

Findings

Expected running time grows cubically with the stability parameter near 0 or 1

Algorithm's expected time is linear in the tempering parameter and initial function value

Numerical results align well with theoretical complexity bounds

Abstract

We construct a fast exact algorithm for the simulation of the first-passage time, jointly with the undershoot and overshoot, of a tempered stable subordinator over an arbitrary non-increasing absolutely continuous function. We prove that the running time of our algorithm has finite exponential moments and provide bounds on its expected running time with explicit dependence on the characteristics of the process and the initial value of the function. The expected running time grows at most cubically in the stability parameter (as it approaches either $0$ or $1$) and is linear in the tempering parameter and the initial value of the function. Numerical performance, based on the implementation in the dedicated GitHub repository, exhibits a good agreement with our theoretical bounds. We provide numerical examples to illustrate the performance of our algorithm in Monte Carlo estimation.