Simple differential equation and Integral Ito

Question

With stochastic differential equation dx(t) = dW (t), and knowing that all integrals occurring are integral Ito. Witch variable changes y = tx. How I can prove?

integral between 0 and t[sdW(s)] = tW(t) - [integral between 0 and t[sdW(s)] W(s) ds]

thanks

Answer 1

Let $X(t)$ be the process governed by the Stochastic Differential Equation

$$dX(t)=dW(t)$$

where $W$ is a Wiener process. If $Y(t)=tX(t)$, then by Ito's Lemma, we have

$$\begin{align} d(tW(t))&=W(t)dt+tdX(t)\\\\ &=W(t)dt+tdW(t) \tag 1 \end{align}$$

Integrating $(1)$ yields

$$\begin{align} \int_0^t d(sW(s))&=tW(t)\\\\ &=\int_0^t W(t)dt+\int_0^tsdW(s)\\\\ \end{align}$$