First passage time problem for the Feller process: numerical analysis of the cumulants approach

The First Passage Time is a random variable that represents the first time a stochastic process enters a certain region or reaches a threshold. The study of the distribution of these variables is generally a difficult problem, both from a theoretical and an applied perspective. First Passage Time problems are very common in many areas of mathematics and applied sciences, especially in the field of neurosciences and mathematical finance. This thesis will discuss the problem of computing the probability density function of the First Passage Time through a constant boundary of a Feller process, a diffusion process very popular in finance and biology. Like most diffusion processes, we don't known the probability density function of the First Passage Time variable (except in a very limited number of cases which are not of interest in the applications), but only its Laplace transform, a very powerful tool allowing (at least theoretically) all the moments to be recovered. Some efforts have been made in the literature to obtain the moments differentiating the Laplace transform, but the expressions obtained were analytically intractable and lead only to the first $3$ moments. On the other hand, the inverse of the Laplace transform is not a known function. A recent algorithm based on cumulants, polynomial transformations of the moments, and on the Laguerre-Gamma polynomial expansion, has been proposed. The main advantage is that, contrary to what happens with moments, closed formulas are available for the cumulants, and this allows to express the First Passage Time probability density function as an infinite series depending on the moments. The generality of the method suggests that it can be applied whenever the Laplace transform can be written as a ratio of functions admitting formal power series expansion. Finally this algorithm is much more efficient than a traditional simulation scheme and allows to make parameter estimation. However, although everything seems fine from a theoretical point of view, a practical problem is to find an optimal number of cumulants in order to minimize the numerical errors arising from the implementation of the iterative algorithm. The aim of this work is to find a stopping criterion that will achieve the best possible approximation, providing improvements and alternatives of the proposed algorithm and assessing its strengths and weakness, mainly through a comparison with simulated data. Moreover some methodologies for an exact simulation of the process are investigated.