On approximating the first passage time density using generalised Laguerre polynomials

Martini Tommaso, University of Turin

The first-passage-time (FPT) problem emerges in various applications where a stochastic process evolves in the presence of a threshold. The mathematical investigation of the FPT problem involves determining the probability density function (pdf) of the FPT random variable. Different strategies exist to address this problem, and their effectiveness relies on formulating an appropriate model for the stochastic process and understanding its properties. Nevertheless, in the majority of the cases a closed-form expression of the FPT pdf is not available, and suitable approximations are needed. This contribution presents a general method that belongs with the broad spectrum of techniques aimed at approximating the FPT density through a constant boundary. The method relies on a Laguerre-Gamma polynomial approximation and iteratively looks for the best degree of the resulting approximating polynomial, until a certain stopping criteria is reached. The latter, coupled with a pair of theoretically justified correction procedures, is also aimed at preserving the positivity of the approximation, a well known issue arising when approximating with a polynomial a pdf supported on the positive half line. The proposed iterative algorithm relies on simple and new recursion formulae involving FPT moments. These moments can be computed recursively from cumulants, if they are known. To provide an example of the procedure in the case of known cumulants of the FPT random variable, the feasibility and effectiveness of the method will be shown when applied to the FPT pdf of a Feller stochastic process, which is not known in a closed form. In this same context, a novel and interesting application consisting in the development of an acceptance-rejection type algorithm will be presented. The latter exploits the functional form of the approximating polynomial and will be theoretically and empirically justified. It will also be shown that the proposed method turns to be particularly useful if only sample data are available, more specifically when a random sample of FPTs is analyzed without any prior information on the stochastic dynamics generating the data. Indeed, in many applications the only available data consist in the direct observation of the FPT variable. This can happen, for instance, in the case of computational neuroscience where, in the popular leaky integrate and fire models, a stochastic process is used to describe the time evolution of the voltage across the neuronal membrane.

Area: IS15 - Stochastic processes in the natural sciences (Giuseppe D'Onofrio/ Serena Spina)

Keywords: First passage time density; Laguerre-Gamma polynomial approximation; cumulants; Feller Process; acceptance-rejection; computational neuroscience;

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