On the Geometric Brownian Motion with Alternating Trend

Antonio Di Crescenzo (), Barbara Martinucci () and Shelemyahu Zacks ()
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Antonio Di Crescenzo: University of Salerno, Department of Mathematics
Barbara Martinucci: University of Salerno, Department of Mathematics
Shelemyahu Zacks: Binghamton University, Department of Mathematical Sciences

A chapter in Mathematical and Statistical Methods for Actuarial Sciences and Finance, 2014, pp 81-85 from Springer

Abstract: Abstract A basic model in mathematical finance theory is the celebrated geometric Brownian motion. Moreover, the geometric telegraph process is a simpler model to describe the alternating dynamics of the price of risky assets. In this note we consider a more general stochastic process that combines the characteristics of such two models. Precisely, we deal with a geometric Brownian motion with alternating trend. It is defined as the exponential of a standard Brownian motion whose drift alternates randomly between a positive and a negative value according to a generalized telegraph process. We express the probability law of this process as a suitable mixture of Gaussian densities, where the weighting measure is the probability law of the occupation time of the underlying telegraph process.

Keywords: Alternating counting process; Exponential random times; Occupation time; Telegraph process (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-05014-0_19

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DOI: 10.1007/978-3-319-05014-0_19

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