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Math Help - Jump-diffusion mean

  1. #1
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    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.

    Can someone please help me?
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  2. #2
    hpe
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    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}.
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