On the distribution of a randomly discounted compound Poisson process (Q1915839)

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.