# Expectation of the product of two Brownian motions [closed]

Could you please let me know the steps to follow to get to the solution?

## closed as off-topic by Daneel Olivaw, LocalVolatility, skoestlmeier, JejeBelfort, HelinNov 23 '18 at 7:36

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• What have you tried? You are supposed to show your own work so we can help you instead of just doing your homework. – Forgottenscience Nov 22 '18 at 12:36

## 1 Answer

I will have a crack at a) i) for you, assuming $$E[W_1] = E[W_2]=0$$:

$$$$\begin{split} E[B_1 W_2] & = E[\alpha W_1 W_2 + \sqrt{1-\alpha^2} W_2 W_2] \\ & = \alpha E[W_1 W_2] + \sqrt{1-\alpha^2}E[W_2^2] \quad \text{(inearity of expectation)}\\ & = \alpha E[W_1] E[W_2] + \sqrt{1-\alpha^2}E[W_2^2] \quad \text{(independence)}\\ & = \sqrt{1-\alpha^2} Var(W_2) \quad \text{(definition of variance)} \end{split}$$$$

Perhaps this helps you with ii)