# Distribution of stochastic integral

I would like to find the distributions of the following random variables:

$Z_k= \frac{1}{\pi} \int^{2\pi}_{0} cos(kt) dW_t$ $k=1,2,...$ and $(W_t)_{t\geq 0}$ is a Wiener process.

What is the distribution of $Z_1$, and $(Z_k)$?

I am new to stochastic calculus, I only know how to integrate a Wiener process wrt. an other Wiener process.

Can someone help me, how to do this?

• Is this a home work? – Aksakal Mar 10 '15 at 16:56
• @Aksakal No, it is not homework. – FelB Mar 10 '15 at 17:25

All you need is to get the variance. Use the fact that $dW_t$ are independent of each other and $(dW_t)^2=dt$ to calculate the variance, which is a square of the integral remembering that the integral is essentially a sum.
• I tried to calculate the variance, I got this: $VAR(Z_1)=\frac{1}{\pi^2}\sum^{n-1}_{0} cos^2t\cdot(t_{j+1}-t_j)=\frac{1}{\pi^2}\int^{2\pi}_{0}cos^2t \ dt =\frac{1}{\pi}$ Is that correct? – FelB Mar 10 '15 at 18:15