Calculate the stochastic integral $\int_0^T W_tdt $

Question

Let $W_t$ be the standard 1-dimensional Brownian motion. Calculate the stochastic integral $$\int_0^T W_tdt $$

Here is my work: Let $ Y_t = U(t, W_t) = t \dot W_t$.

Applying Ito lemma, $dY_t = W_tdt + tdW_t$

Taking integral, $Y_T-Y_0 = \int_0^TW_td_t+\int_0^TtdW_t$

But how to find the value of $\int_0^TtdW_t$? Or is there any other $Y_t$ that I should take?

Answer 1

$dW_t$ and $dW_s$ are independent increments and there is no relationship between them. Hence, $\int_0^T t dW_t$ can not be integrated or “reduced”.

In fact, the variable defined by $\int_0^T tdW_t$ is itself a random variable, which has its own mean, variance, etc.