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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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I think the object you really want to be integrating is a Gaussian process with some specified mean and covariance function. And you'll probably want the covariance function to have some regularity to it (e.g. continuous) in order to guarantee that the integral exists. In this case the result of the integral will be normally distributed, so you only need to compute its mean and variance. This is straightforward using Fubini's theorem. See also my answer here.