On the distribution of a randomly discounted compound Poisson process
type: Item
instance of: scholarly article
title: On the distribution of a randomly discounted compound Poisson process
zbMATH Open document ID: 0853.60048
DOI: 10.1016/0304-4149(95)00076-3
author: Trygve Nilsen
author: Jostein Paulsen
published in: Stochastic Processes and their Applications
publication date: 1996-08-05T00:00:00Z
review text: The distribution of the stochastic integral \(Z= \int^\infty_0 e^{-R_t} dP_t\) is studied. Here \(R_t\) is a Brownian motion with positive drift and \(P_t\) is an independent compound Poisson process, i.e. \(P_t= \sum^{N_t}_{i=1} S_i\) where \(N_t\) is a Poisson process and \(\{S_i\}\) a sequence of i.i.d. nonnegative random variables, independent of \(N_t\). Let \(\{T_i\}\) be the times of jumps of the process \(N_t\), the stochastic integral \(Z\) can be written as \(Z= \sum^\infty_{i=1} S_i e^{-R_{T_i}}\). The main result is to show that with exponentially distributed jumps, \(Z\) has the same distribution as that of a gamma distributed random variable divided by an independent beta distributed random variable.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 894801
zbMATH Keywords: compound Poisson process
zbMATH Keywords: Brownian motion
reviewed by: Viktor Ohanyan
MaRDI profile type: Publication
full work available at URL: 0304-4149(95)00076-3
OpenAlex ID: W2019352205
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description: scientific article; zbMATH DE number 894801