Fast Exact Simulation of the First Passage of a Tempered Stable Subordinator Across a Non-Increasing Function

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, nonincreasing, 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 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. Funding: J. I. González Cázares and A. Mijatović are supported by the EPSRC Grant EP/V009478/1 and by The Alan Turing Institute under the EPSRC grant EP/X03870X/1. A. Mijatović is also supported by the EPSRC grant EP/W006227/1. F. Lin is funded by The China Scholarship Council and The University of Warwick PhD scholarship.