Moment Generating Function of a summation of random variables where the upper limit is also random

Question

How do we compute the Moment Generating Function of Q(t) here? I understand that we can use Wald's Equation to compute E[Q(t)]. Is there any theorem which can help me solve for the Moment Generating Function of Q(t)?

Answer 1

First show that $\begin{eqnarray*}\mathbb{E}\left(e^{uQ(t)}\Big|N(t)\right)&=&\left(\varphi_{Y}(u)\right)^{N(t)}\end{eqnarray*}$. Then, enforce total law of expectation: $$\begin{eqnarray*}\varphi_{Q(t)}(u)&=&\mathbb{E}\left(e^{uQ(t)}\right) \\ &=& \mathbb{E}\left(\mathbb{E}\left(e^{uQ(t)}\Big|N(t)\right)\right) \\ &=&\mathbb{E}\left(\left(\varphi_{Y}(u)\right)^{N(t)}\right) \\ &=& \sum_{n=0}^{\infty}\left(\varphi_{Y}(u)\right)^n\mathbb{P}\left(N(t)=n\right) \\ &=& e^{\lambda t \left(\varphi_{Y}(u)-1\right)}\end{eqnarray*}$$