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How can I write the following expectation $E[f(X_t,X_s)]$ by means of a Lebesgue integral and the density of $X_t$? where $f$ is a "nice" function and $X_t$ is a process without independent increments!

Is there a way to do it without using joint distributions if we know the quantity $E[(X_t-X_s)X_s]$ for $t>s$?

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    Unless $f(x,y)$ is of the form $a(x) + b(y) + c x y$, there's not much hope of doing it without knowing something about the joint distribution.2012-10-25
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    @RobertIsrael: $f$ is of the form $f(X_t,X_s)=f(X_t)f(X_s)$ I mean, it is in fact a product of functions.2012-10-29

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