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I need to define a new process from for example Wiener process like $U(t)=f(W(t))$, (for example $f(x)=1+x^2$ ) and then calculate the average like $\langle U(t)U(s)\rangle$. Is it possible?

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yes for example simply raising the processes to powers does not work: WienerProcess[]^2 – richard Jun 2 '14 at 13:31
Comment not an answer, in the documentation pages for the forthcoming Wolfram Language release (and the version of M10 on Raspberry Pi) you can see there is such functionality coming - – Martin John Hadley Jun 4 '14 at 21:00
@MartinJohnHadley, nice, thanks. – richard Jun 7 '14 at 9:16
up vote 2 down vote accepted

Possibly an overkill for this case but quite general :

define a simple process which returns the variable you want

procU = ItoProcess[\[DifferentialD]x[t] == \[DifferentialD]w[t],  1 + x[t]^2, 
           {x, 0}, {t, 0}, w \[Distributed] WienerProcess[]] ;

now you can use it as :

(* 1 + t *)

CovarianceFunction[procU, s, t]
(* 2 Min[s, t]^2 *)
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