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

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.

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On the Probabilities of Correlated Defaults: a First Passage Time Approach

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2008.13.1.14593 ◽

2008 ◽

Vol 13 (1) ◽

pp. 117-133 ◽

Cited By ~ 4

Author(s):

M. Valužis

Keyword(s):

Brownian Motion ◽

Credit Risk ◽

First Passage Time ◽

Joint Probability ◽

Passage Time ◽

Geometric Brownian Motion ◽

First Passage ◽

Analytical Expression ◽

Exact Analytical Expression ◽

Correlated Defaults

This article investigates the joint probability of correlated defaults in the first passage time approach of credit risk subject to condition that the underlying firms’ assets values and the default boundaries follow geometric Brownian motion processes. The exact analytical expression of joint probability of two correlated defaults in the case of stochastic default boundaries is presented. Also, some properties of this solution are provided.

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On the First Passage time for Brownian Motion Subordinated by a Lévy Process

Journal of Applied Probability ◽

10.1239/jap/1238592124 ◽

2009 ◽

Vol 46 (1) ◽

pp. 181-198 ◽

Cited By ~ 9

Author(s):

T. R. Hurd ◽

A. Kuznetsov

Keyword(s):

Brownian Motion ◽

Approximation Scheme ◽

First Passage Time ◽

Gaussian Model ◽

Passage Time ◽

Financial Mathematics ◽

First Passage ◽

Financial Modeling ◽

Motion Time ◽

Normal Inverse Gaussian

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

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On the area under a continuous time Brownian motion till its first-passage time

Journal of Physics A General Physics ◽

10.1088/0305-4470/38/19/004 ◽

2005 ◽

Vol 38 (19) ◽

pp. 4097-4104 ◽

Cited By ~ 43

Author(s):

Michael J Kearney ◽

Satya N Majumdar

Keyword(s):

Brownian Motion ◽

Continuous Time ◽

First Passage Time ◽

Passage Time ◽

First Passage

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Occupation Times for Markov-Modulated Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200009268 ◽

2012 ◽

Vol 49 (02) ◽

pp. 549-565 ◽

Cited By ~ 4

Author(s):

Lothar Breuer

Keyword(s):

Brownian Motion ◽

First Passage Time ◽

Passage Time ◽

Laplace Transforms ◽

Occupation Times ◽

First Passage ◽

Scale Functions ◽

Positive Variation ◽

Markov Modulated

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

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