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What can we say about distribution of

$\int_t^TN(\mu(s),\sigma^2(s))ds$

,where $N(\mu,\sigma)$ is independent normally distributed with mean $\mu(s)$ and variance $\sigma^2(s)$, $T$ and $t$ are finite?

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    To even define the integral $\int\limits_t^TX_sds$ where $X_s$ has gaussian distribution $(\mu(s),\sigma^2(s))$, one needs to specify the joint distribution of the family $(X_s)_{t\leqslant s\leqslant T}$.2012-10-27
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    @did Yes, I forgot this. Let's say they are iid.2012-10-27
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    Then the integral will not be defined.2012-10-27
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    They can't be iid; you're explicitly saying that they have different distributions (different means and variances) -- @did you mean "independently distributed", as the question now reads?2012-10-27
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    @joriki Yes, sorry. I meant just indepedent.2012-10-27
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    @did Yes, it seems so. What kind of assumptions required to make it well defined? I can add this to a question if this makes it meaningful.2012-10-27
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    @joriki If the random variables are independent (and with positive variance), the integral cannot be defined pathwise.2012-10-28
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    @did: Sorry, that was just a pun on your new pseudonym :-) I was actually responding to learningmath's comment.2012-10-28
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    @joriki Sorry to be a bit... slow. :-)2012-10-28

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