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How can I write the folowing 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 undependent 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 $t>s$?

Thanks a lot! :)

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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. –  Robert Israel Oct 25 '12 at 19:10
    
@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. –  user46470 Oct 29 '12 at 19:57

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