A Monte Carlo method for the simulation of first passage times of diffusion processes
The authors consider a diffusion process satisfying a one-dimensional time-homogeneous stochastic differential equation of Itô-type. They are interested in determining via a numerical approximation the first passage time, that is the first time the diffusion process reaches an absorbing boundary. The proposed numerical method is a nested Monte Carlo algorithm in the following sense. The outer ``loop'' consists of simulating (with a strong scheme such as the Milstein scheme) \(N\) trajectories of the process on a grid on the interval of interest. If a trajectory has passed the boundary on one of the subintervals of the grid, its first passage time is estimated to be the middle of the subinterval, and a new trajectory is started. The inner ``loop'' on each subinterval is required to account for undetected crossings that may occur inside it. A suitable tied-down process is introduced and the probability of it crossing the boundary is determined using a Monte Carlo method on the tied-down process. An analysis of the algorithm is carried out and the advantages of the new method compared with previously proposed methods are discussed. Finally, numerical examples are presented.
scientific article; zbMATH DE number 1784512