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Math Help - change of measure martingale

  1. #1
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    change of measure martingale

    Hallo!

    There's a given probability space (\Omega,\mathcal{F},P) equipped with a filtration (\mathcal{G}_t)_{t \in [0,T]} and a subfiltration \mathcal{F}_t \subset \mathcal{G}_t.


    And now I want to do a change of measure

    \frac{dQ}{dP}|_{\mathcal{F}_t}= Z_t, with

    Z_t=\exp\{-\int\limits_0^t\frac{\mu_t -r}{\sigma}dW_t-\frac{1}{2}\int\limits_0^t(\frac{\mu_t -r}{\sigma})^2dt\}


    where
    \mu_t:=E[\mu|\mathcal{F}_t] is an estimator for the unknown parameter \mu which has a normal prior distribution \mathcal{N}(\mu_0,{\sigma_0}^2)
    and W is a brownian motion with respect to \mathcal{F}_t
    r,\sigma >0 are constants
    t \in [0,T]

    I've also an explicit version of \mu_t

    \mu_t=\frac{\sigma^2\mu_0+{\sigma_0}^2(\mu t+\sigma V_t)}{\sigma^2 + {\sigma_0}^2t}
    V is a brownian motion with respect to \mathcal{G}_t and \mu is independent of V


    How can I now prove, that Z_t is a \mathcal{F}_t-martingale?


    I only know the Novikov condition,
    E[exp\{\frac{1}{2}\int\limits_0^T(\frac{\mu_t -r}{\sigma})^2dt \}] < \infty.
    But I think this condition isn't fulfilled.


    Can anybody help me?
    Thanks in advance!
    Last edited by Juju; March 30th 2011 at 04:04 AM.
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