On the computation of first-passage-times by
means of simulation techniques
M.T. Giraudo and L. Sacerdote
Department of Mathematics, University of Torino,
V.C. Alberto 10, 10123 Torino, Italy
In a large number of problems of biological, physical, engineering or ecological interest the necessity arises to evaluate first-passage-time distributions for stochastic processes. In particular a wide literature exists on this subject (cf. [1], [8], [5], [11], [7], [4] and references quoted therein) in which analytical or numerical techniques are proposed in order to obtain evaluations of the moments or of the asymptotical behaviour of first-passage-time distributions for diffusion processes through a boundary ([2], [6], [3], [9] and references quoted therein). However in various complex instances none of these techniques can be applied in order to obtain quantitative results of applicational interest. For example, even in the simple case of an Ornstein-Uhlenbeck diffusion it is not possible to determine analytical expresions for the moments of the first-passage-time through a time depending boundary. This is the reason why in various instances researchers largely use simulations to investigate on such problems. Unluckily, while several theorems can be proved on convergence of simulated realizzations to the real ones, no convergence result holds for the simulated first-passage-time. The problem is not of easy solution since the discretization of the process causes the loss of those cases in which the first passage would happen during the interval step. In this way the simulated process sistematically underestimates the values of first-passage-times. Here we compare simulation and analytical results by means of some applicational interesting examples with the idea to throw some light on the magnitude order of errors connected with simulations. Furthermore we attempt to improve the simulation algorithm suggested by [10] making use of the connection between the first-passage-time distribution for the considered diffusion process and that for the corresponding tied-down diffusion process.