How to use Itō in this very simple case

Question

I want to apply Ito for the following process:
\begin{equation*} X_t = tW_t + \int_0^t W_u du, \end{equation*} where $W$ is a Brownian motion. I have no trouble with the part $tW_t$ This can be written $tW_t = f(t,W_t)$ with $f(t,x) = tx$. However, which function is used for the integral? Can I write $X_t = g(t,W)$ but how is the integral a function of $t$ and $x$?

Answer 1

You can solve this problem using Ito's Lemma two times. The first time is by the multidimensional version: if $A_t = B_t + C_t$ then $dA_t = dB_t + dC_t$ (Linearity).

In your example: $$d(X_t) = d(tW_t) + d\left(\int_0^tW_udu\right)$$ Next you apply Ito to the first component (Product Rule) $$d(tW_t) = W_tdt + tdW_t$$ and the second one is the definition of a time integral $$d\left(\int_0^tW_udu\right) = W_t dt$$ So the answer is $$dX_t = 2W_tdt+tdW_t$$