Orthogonal gamma-based expansion for the CIR's first passage time distribution

In this paper we analyze a method for approximating the first-passage time density and the corresponding distribution function for a CIR process. This approximation is obtained by truncating a series expansion involving the generalized Laguerre polynomials and the gamma probability density. The suggested approach involves a number of numerical issues which depend strongly on the coefficient of variation of the first passage time random variable. These issues are examined and solutions are proposed also involving the first passage time distribution function. Numerical results and comparisons with alternative approximation methods show the strengths and weaknesses of the proposed method. A general acceptance-rejection-like procedure, that makes use of the approximation, is presented. It allows the generation of first passage time data, even if its distribution is unknown.