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More explicitly, if $W(t)$ is Brownian motion, what would be $$f(t) := \int_0^t u dW(u)$$ and $$g(t) := \int_0^t W(u) du$$?

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  • $\begingroup$ $f + g = tW$ by application of Ito. $\endgroup$
    – user34971
    Commented Dec 7, 2021 at 11:43
  • $\begingroup$ @FridoRolloos indeed, but i'm interested in $f$ and $g$ separately. $\endgroup$
    – athos
    Commented Dec 7, 2021 at 11:43
  • $\begingroup$ I do not think there is an explicit expression for the separate terms. $\endgroup$
    – user34971
    Commented Dec 7, 2021 at 11:44
  • $\begingroup$ @FridoRolloos mmm.. is there a criterion on which integrals have or have not explicit expressions? $\endgroup$
    – athos
    Commented Dec 7, 2021 at 11:51
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    $\begingroup$ For every $t \geq 0,$ both $f(t)$ and $g(t)$ are random variables $N(0,t^3/3).$ $\endgroup$
    – Sebastian
    Commented Dec 7, 2021 at 12:11

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