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On the distribution of Verhulst process

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Abstract

We investigate a Verhulst process, which is a particular functional of geometric Brownian motion and has many applications, among others, in biology and in stochastic volatility models. We present a representation of the density of one-dimensional distribution of Verhulst process. The closed formula for the density of Verhulst process simplifies in the case where the drift of the geometric Brownian motion is equal to −1/2. Some special properties of this process are discussed; in particular, it turns out that, under Girsanov’s change of measure, a Verhulst process still remains a Verhulst process, although with other parameters.

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, Banacha str. 2, 02-097, Warszawa, Poland

    Jacek Jakubowski & Maciej Wiśniewolski

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  1. Jacek Jakubowski
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  2. Maciej Wiśniewolski
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Correspondence to Jacek Jakubowski.

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Jakubowski, J., Wiśniewolski, M. On the distribution of Verhulst process. Lith Math J 55, 91–101 (2015). https://doi.org/10.1007/s10986-015-9267-y

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  • DOI: https://doi.org/10.1007/s10986-015-9267-y

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