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I wish to express as a Lebesgue integral the following expectation,

$E[\varphi(B_t)\varphi(B_s)]=\int ?$

for $0\leq s\leq t$, where $B_t$ is a Brownian motion with law $N(0,\sigma^2 t)$. Any ideas? I guess the point is to use independent increments since I would like to avoid the joint distribution.

Thank you very much! :)

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    What is $\varphi$?2012-10-21
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    A function/transformation. Like when you do: $E[\varphi(X)]=\int_{\mathbb{R}} \varphi(x)f(x)dx$ where $f$ is the density of $X$.2012-10-21
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    But, I meant, are there conditions on this function or not? I guess at least for example measurable bounded, or something like that. We can try to compute a density of $(B_t,B_s)$ as we know those of $(B_t-B_s,B_s)$.2012-10-21
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    Yes of course, put whatever conditions you need on $\varphi$ but yes, must be at least measurable so that the expectation makes sense.2012-10-21
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    @Daniel: your accounts have been merged. In the future please register your account to avoid accidentally creating duplicates.2012-10-23

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