Expectation of a Product of Correlated Brownian Motions

Summary

The document gives a short derivation for the expectation of the product of two Brownian-related random variables. It represents one variable as a linear combination of an independent Brownian motion and another Brownian motion, then expands the product and applies linearity of expectation. Under the stated assumption that the underlying variables have zero means and are independent, the cross term vanishes, leaving a term proportional to the variance of the second motion.

This is a narrow probability calculation that can help with covariance reasoning in stochastic models. The answer addresses only one requested subpart and offers no discussion of time-indexed Brownian variances, correlation structure beyond the construction, or how to proceed with the remaining part of the original exercise. Its result depends on the assumptions stated in the derivation; different dependence or mean assumptions would change the calculation.

Key ideas

• Expand the product after expressing one Brownian variable as a linear combination of independent terms.
• Linearity of expectation separates the resulting terms.
• Independence and zero means make the cross expectation vanish.
• The remaining contribution is determined by the variance of the second Brownian variable.
• The derivation covers only a limited part of the underlying problem.

Tags

# Expectation of the product of two Brownian motions


# Expectation of the product of two Brownian motions












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

## Answer by Attack68 (score 1, accepted)

https://quant.stackexchange.com/a/42729

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

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

Perhaps this helps you with ii)

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