Currently I am working on studying stochastic integrals of the form: $$Z\_\infty = \int\_0^\infty e^{-f(t)}\mathop{d}S\_t$$


where $S\_t$ is a Compound-Poisson process with Exponentially-distributed increments. We know that the integral converges so long as $f$ is continuous on $[0,\infty)$ and $\lim\_{t\to\infty}f(t) \to \infty$ holds true. $Z\_\infty$ effectively models a perpetuity where $f$ is our discount rate function, and $S\_t$ is the cashflow aggregation process.


As far as I am aware, in the case where $f$ is linear, $Z\_\infty$ is Gamma distributed. I am interested in knowing if there are further results for when the discounting is a power-law rate; $f(t) = \lambda t^\alpha $.



Are there any known results on the probability distributions of perpetuities with power law discount rates?
This is more of an extended comment trying to study this integral.


In the similar spirit as [here](https://math.stackexchange.com/questions/2586336/stochastic-integral-of-poisson-process/2587999#2587999) we study the Laplace transform. As explained [here](https://personal.ntu.edu.sg/nprivault/MA5182/stochastic-calculus-jump-processes.pdf) we have the Lévy-Khintchine formula for compound Poisson processes for a process whose increments have distribution $\nu$:


$$E(exp(r\int\_{0}^{T}f(t)dY\_{t}))=exp(\lambda\int\_{0}^{T}\int\_{-\infty}^{\infty}(e^{yrf(t)}-1) \nu(dy) dt )$$


and so in the above case of $\nu(y)=e^{-\mu y}1\_{y\geq 0}$ and $f(t)=exp(-t^{\alpha})$ we have
$$exp(\lambda\int\_{0}^{T}\int\_{0}^{\infty}(e^{yr exp(-t^{\alpha})}-1) e^{-\mu y} dt ).$$


I tried in Mathematica to possibly get exact formulas for the above for general $a>1$ and $T=\infty$ but didn't get anything back. It might require (keyhole) contour methods to study. You could also try to directly invert its Laplace transform but that seems tricky.


