Let $\tau$ be an exponential random variable, with parameter $\lambda$. Let
$$
V = \delta^\tau
$$
where $0 < \delta <1$. Sorry if this notation seems strange, but it is what I am using, I will get confused if I switch it around! What I want is the moment generating function of $V$. The density function of $\tau$ is $f(\tau) = \lambda \exp(-\lambda \tau)$. The change of variable to $V$ is
$$
\delta^\tau = V \\
\tau \log \delta = \log V \\
\tau = \frac{\log V}{\log \delta}
$$
And $\frac{\partial \tau }{\partial V } = \frac{1}{V \log \delta}$. So the density of $V$ is
$$
g(v) = f(\frac{\log v}{\log \delta}) \frac{\partial \tau }{\partial V }\\
= \lambda \exp(-\lambda \frac{\log v}{\log \delta})\frac{1}{v \log \delta}
$$
The moment generating function of $V$ is therefore
$$
MGF_V(t) = \int_0^1 \exp(t v) \lambda \exp(-\lambda \frac{\log v}{\log \delta})\frac{1}{v \log \delta} dv
$$
Integrating this is Mathematica gives me a seriously complicated function of Gamma functions:
$$
-\frac{\lambda}{\log \delta} (-t)^{\frac{\lambda}{\log \delta}} (\Gamma(-\frac{\lambda}{\log \delta})-\Gamma(\frac{-\lambda}{\log \delta} ,-t))
$$
where the second gamma is using Mathematica's implementation of the Incomplete Gamma function. I need to do further things with this function, including use it numerically in either Matlab or Mathematica, and have serious trouble working with it. Compare this to the pleasantly simply moment generating function of $\tau$
$$
MGF_{\tau}(t) = \int_0^{\infty} \exp{(t \tau)} \lambda \exp(-\lambda \tau) d \tau = \frac{\lambda}{\lambda – t}
$$
To have a sense of what I'm doing, think of $\tau$ as wait time, and $\delta$ as a discount factor. So if I have wait time 0 I get value 1, but if I have to wait two periods I get value $\delta^2$. Everything that is going on with $V$ is completely pinned down by $\tau$, so I feel like I should be able to work in terms of $\tau$, but I know I can't just perform my change of variable after calculating the MGF. Can I?
For further information, there are a variety of ways to express the $MGF_V(t)$, depending on how you feed it to Mathematica. One additional way is
$$
-\frac{\lambda}{\log \delta} ( – ExpIntegralE[1+\frac{\lambda}{\log \delta},-t]+(-t)^{\frac{\lambda}{\log \delta}} \Gamma[-A])
$$
where ExpIntegralE is Mathematica's implementation of the Exponential Integral Function:
$$ExpIntegralE[n,t] = \int \exp(-z t) \frac{1}{t^n} dt$$
I've done a lot of messing about with various forms of this expression, but none are really helpful. I need, eventually, to take derivatives of this thing and or calculate it numerically. Eventually I have a calculus of variations problem that uses it! All these implementations (including Matlab's, which are annoyingly different that Mathematica's) seem to have real instability issues, at least the way I want to use them.
This is cross posted from Mathematics, didn't get much interest there.
Best Answer
Since $0 < \delta <1$, then $\log \delta <0$. So the density of $V$ is
$$g(v) = f\left(\frac{\log v}{\log \delta}\right)\cdot \left|\frac{\partial \tau }{\partial V }\right| = \lambda \exp\left\{-\lambda \frac{\log v}{\log \delta}\right\}\frac{1}{v |\log \delta|} \\= \frac {\lambda}{|\log \delta|}\frac 1v\exp\left\{ \frac{\lambda}{|\log \delta|}\log v\right\}$$
Set $\alpha \equiv \frac{\lambda}{|\log \delta|}$. Then, manipulating,
$$g(v) = \alpha v^{\alpha-1}, \;\;v\in [0,1] $$
which is the density of a $\text{Beta}(\alpha,1)$ distribution.
The moment generating function is
$$MGF_V(\alpha,1,t) = 1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+r+1} \right) \frac{t^k}{k!}$$
and which, among other things provides a nice recursive formula
$$E[V^s] = \frac {\alpha +s-1}{\alpha+s}E[V^{s-1}]$$