Mean and Variance of a stochastic process

How do you find the mean and variance for:

$$Z_{T} = \int_{0}^{T} |W_{t}|^{1/2} dW_{t}$$

$$Y_{T} = \int_{0}^{T} \text{sign}(W_{t}) dW_{t}$$ where $$\text{sign}(y) = 1$$ for $$y \ge 0$$ and $$0$$ otherwise.

I'm trying to use Ito's lemma for $$Z_{t}$$ where $$dZ_{t}= f_{t}dt + f_{x}dW_{t} + \frac{1}{2}f_{xx}dt$$ but I don't know what to use for $$f(t, x)$$.

• For mean: use that ("nice") stochastic integrals are martingales; hence, they have constant expectation. For variance: use Itô's isometry – saz yesterday