Are there any known results on the probability distributions of perpetuities with power law discount rates?

Question

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 $.

Answer 1

This is more of an extended comment trying to study this integral.

In the similar spirit as here we study the Laplace transform. As explained here 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.