On the distribution of Verhulst process

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• 作者：Jacek Jakubowski ; Maciej Wi?niewolski
• 关键词：geometric Brownian motion ; Verhulst process ; Girsanov’s change of measure ; Laplace transform ; exponential functional of Brownian motion ; 60J70 ; 60H30 ; 60J65
• 刊名：Lithuanian Mathematical Journal
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：55
• 期：1
• 页码：91-101
• 全文大小：171 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Statistics
  Algebra
  Russian Library of Science
  
• 出版者：Springer New York
• ISSN：1573-8825

文摘

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 ?/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.