Right now I dont know how to find the expectation and the variance for the following stochastic integral:
$$\int\_{0}^t B(s) dW(s)$$
where $B(t)$ and $W(t)$ are correlated standard Brownian Motions with $d\langle B, W \rangle\_t = \gamma dt$.


Usually, when $B(t)$ and $W(t)$ are independent, we know 
$$\mathbb{E}\left(\int\_{0}^t B(s) dW(s)\right) = 0$$ and by Ito's isometry, 
$$Var\left(\int\_{0}^t B(s) dW(s)\right) = \mathbb{E}\left(\int\_{0}^t B(s) dW(s)\right)^2 = \int\_{0}^t \mathbb{E}B(s)^2 ds.$$


Does these results still apply?



How to Calculate the Expectation and Variance for Stochastic Integral with correlated Brownian Motions
Decompose your Brownian motion $W$ as $W\_t = \gamma B\_t + \bar \gamma B\_t^\bot$, where $\bar \gamma = \sqrt{1-\gamma^2}$ and $B^\bot$ is independent from $B$. Then:


$$\int\_0^t B\_s \, \mathrm dW\_s = \gamma \int\_0^t B\_s \, \mathrm dB\_s + \bar\gamma \int\_0^t B\_s \, \mathrm dB\_s^\bot$$


The first term can be computed using Itô's formula, the second one involves two independent Brownian motions. 


