On exact simulations of first hitting times of the solutions of SDEs

Abstract

In this thesis, we look into exact simulations of first hitting times of the solutions to stochastic differential equations with unit diffusion coefficient. While discretization schemes do represent essential tools in SDE simulations, their inherent errors prove them to be unsuitable for fields where precision is necessary. Through our exploration of the work done in Exact Simulation of the First-Passage Time of Diffusions (2019) by S. Herrmann and C. Zucca, we show that it is indeed possible to construct an algorithm which exactly simulates the first hitting time of the solution of a stochastic differential equation. Our key tools are Girsanov’s theorem, which allows us to shift the measure under which we are looking at the problem to put it into a frame which is more suitable for simulations, as well as the acceptance-rejection scheme which was originally proposed in Exact Simulation of Diffusions (2005) by A. Beskos and G. O. Roberts.