Exact simulation of the first time a diffusion process overcomes a given threshold
Abstract:
The aim is to propose a new exact simulation method for the first passage time (FPT) of a diffusion process. We shall consider either a continuous diffusion process (in collaboration with Cristina Zucca, University of Turin) or a jump diffusion (in collaboration with Nicolas Massin, Auch, France). We denote τL the first passage time through the level L. In order to exactly simulate τL, we cannot use an explicit expression of its density. The classical way to overcome this difficulty is to use efficient algorithms for the simulation of sample paths, like discretization schemes. Such methods permit to obtain approximations of the first-passage times as a by-product.
For efficiency reasons, it is particularly challenging to simulate directly this hitting time by avoiding to construct the whole paths. The authors introduce a new rejection sampling algorithm which permits to perform an exact simulation of the first-passage time for general one-dimensional diffusion processes. The main ideas are based both on a previous algorithm pointed out by A. Beskos et G. O. Roberts which uses Girsanov's transformation and on properties of Bessel paths. The efficiency of the method is described through theoretical results and numerical examples.
