On the Cumulants of the First Passage Time of the Inhomogeneous Geometric Brownian Motion

Elvira Di Nardo and Giuseppe D’Onofrio
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Elvira Di Nardo: Dipartimento di Matematica ‘G. Peano’, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Giuseppe D’Onofrio: Dipartimento di Matematica ‘G. Peano’, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Mathematics, 2021, vol. 9, issue 9, 1-17

Abstract: We consider the problem of the first passage time T of an inhomogeneous geometric Brownian motion through a constant threshold, for which only limited results are available in the literature. In the case of a strong positive drift, we get an approximation of the cumulants of T of any order using the algebra of formal power series applied to an asymptotic expansion of its Laplace transform. The interest in the cumulants is due to their connection with moments and the accounting of some statistical properties of the density of T like skewness and kurtosis. Some case studies coming from neuronal modeling with reversal potential and mean reversion models of financial markets show the goodness of the approximation of the first moment of T . However hints on the evaluation of higher order moments are also given, together with considerations on the numerical performance of the method.

Keywords: hitting times; Brennan–Schwartz model; formal power series; multiplicative noise (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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