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 $f(t, x)$.

  • 1
    $\begingroup$ For mean: use that ("nice") stochastic integrals are martingales; hence, they have constant expectation. For variance: use Itô's isometry $\endgroup$ – saz yesterday

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