Jump-diffusion mean

Quote:

Originally Posted by McGoose

I have a problem calculating the mean for the following probability distribution,

exp(sum(X))

where the sum is Poisson-distributed with intensity lamda and X is a normal distribution with mean my and variance V, i.e. the number of normal variables is Poisson distributed.

Let's take $\mu = 0, \, \sigma = 1$ to keeps things simple.
Conditioned on $N$ = number of terms, $Y_N = \sum_{k=1}^N X_k$ has a normal distribution with mean 0 and standard deviation $\sqrt{N}$ . Then $e^{Y_N}$ has mean $e^{N/2}$ (lognormal distribution with location parameter 0). Now take the expectation with respect to N:
$\sum_{n=0}^\infty e^{-\lambda} \frac{\lambda^n e^{n/2}}{n!} = e^{(\sqrt{e}-1)\lambda}$ .