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On the Geometric Brownian Motion with Alternating Trend

In: Mathematical and Statistical Methods for Actuarial Sciences and Finance

Author

Listed:
  • 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)

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.

Suggested Citation

  • Antonio Di Crescenzo & Barbara Martinucci & Shelemyahu Zacks, 2014. "On the Geometric Brownian Motion with Alternating Trend," Springer Books, in: Cira Perna & Marilena Sibillo (ed.), Mathematical and Statistical Methods for Actuarial Sciences and Finance, edition 127, pages 81-85, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-05014-0_19
    DOI: 10.1007/978-3-319-05014-0_19
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