Computing the Expected Product of Brownian Motion Integrals

Summary

The document asks how to find the expectation of the product of two integrals of standard Brownian motion over the unit interval, one weighted by time squared. The response evaluates the product as a double integral and moves the expectation inside the integral, using the covariance of Brownian motion at two times.

That covariance is the smaller of the two time points. Splitting the inner integral at the point where the ordering changes turns the calculation into ordinary integration. The document gives the setup and reduction but leaves the final arithmetic to the reader. It offers a compact example of using covariance structure and integration to calculate a moment; it does not discuss financial applications, assumptions beyond Brownian motion, or extensions to other processes.

Key ideas

• The expectation of a product of Brownian motion integrals can be expressed as a double integral of covariances.
• Brownian motion values at two times have covariance equal to the earlier time.
• The integration region can be split where the two time variables exchange order.
• The response provides the integral setup but leaves its final evaluation unstated.

Tags

# For the Brownian motion integrate


# For the Brownian motion integrate












I want to calculate $$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$

I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to get the other one and the full answer of their multiplied expectation.

## Answer by Gordon (score 7)

https://quant.stackexchange.com/a/42977

Note that \begin{align*} E\left(\int_0^1 W_t\, dt \int_0^1 t^2W_t\, dt \right) &= E\left(\int_0^1\!\!\!\int_0^1 s^2 W_s W_t\, dsdt \right)\\ &=\int_0^1\!\!\!\int_0^1 s^2 E(W_s W_t)\, dsdt\\ &=\int_0^1\!\!\!\int_0^1 s^2 (s\wedge t)\, dsdt\\ &=\int_0^1 s^2\,ds \int_0^1 s\wedge t\, dt\\ &=\int_0^1 s^2\,ds \left(\int_0^s t\,dt + \int_s^1 s\,dt\right). \end{align*} The remaining is now straightforward.

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