Jump-diffusion mean

• November 30th 2005, 04:01 AM
McGoose
Jump-diffusion mean
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.

• December 8th 2005, 04:48 PM
hpe
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:
$\sum_{n=0}^\infty e^{-\lambda} \frac{\lambda^n e^{n/2}}{n!} = e^{(\sqrt{e}-1)\lambda}$.