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.
Highlights
Mathematics 2021, 9, 956. https://Stochastic diffusion processes with linear drift and multiplicative noise are often considered both in theory and applications because they constitute a good compromise between adherence to reality and mathematical tractability when used as models of real phenomena
Much is known about this process, a lack of results emerges when dealing with its version with non-zero asymptotic mean, namely the Inhomogeneous Geometric Brownian Motion (IGBM)
In the interest rates field, it is called the Brennan-Schwartz model [3,4], denoted as the GARCH model when used for stochastic volatility and for energy markets [5], as Lognormal diffusion process with exogenous factors when used for forecasting and analysis of growth [6,7], in real option literature, it goes under the names of Geometric
Summary
Stochastic diffusion processes with linear drift and multiplicative noise are often considered both in theory and applications because they constitute a good compromise between adherence to reality and mathematical tractability when used as models of real phenomena. The starting point is the availability of a closed-form expression of g∗ (z) in terms of power series This is not the case for the IGBM, but we exploit a property of the Tricomi function involving an asymptotic expansion. The second application deals with models of financial markets with mean reversion where the study of extreme events, such as defaults, can be studied using the problems of first-passage time or exit time from a region In this context the IGBM is better know as the GARCH diffusion process or Brennan–Schwartz model and it is characterized by the properties that the changes in the short rate are state-dependent and unlimited excursions of the process are not allowed (see for instance [3,4,28])
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