A cumulant approach for the first-passage-time problem of the Feller square-root process

TL;DR

This paper introduces a cumulant-based approximation method for the first passage time density of the Feller process, leveraging polynomial approximations and symbolic calculus for improved accuracy in neuronal and financial applications.

Contribution

It develops a novel cumulant and polynomial approximation approach for the first passage time problem of the Feller process, with explicit formulas and implementation guidance.

Findings

Accurate approximation of first passage time densities in neuronal and financial models.

Method performs well even with few polynomial terms.

Provides conditions to enhance approximation accuracy.

Abstract

The paper focuses on an approximation of the first passage time probability density function of a Feller stochastic process by using cumulants and a Laguerre-Gamma polynomial approximation. The feasibility of the method relies on closed form formulae for cumulants and moments recovered from the Laplace transform of the probability density function and using the algebra of formal power series. To improve the approximation, sufficient conditions on cumulants are stated. The resulting procedure is made easier to implement by the symbolic calculus and a rational choice of the polynomial degree depending on skewness, kurtosis and hyperskewness. Some case-studies coming from neuronal and financial fields show the goodness of the approximation even for a low number of terms. Open problems are addressed at the end of the paper.