Moments of the Time Integral of Squared Brownian Motion

Summary

The document addresses whether the time integral of squared Brownian motion is a random variable and how to find its first two moments. It computes the expectation by interchanging expectation and integration, then using the fact that standard Brownian motion at time s has second moment s. Integrating this over time from zero to t gives an expected value of one half times t squared.

For the variance, the document points to Itô’s isometry and an external derivation rather than providing the calculation. It therefore offers only a partial treatment of the requested moments and does not spell out the measurability argument that establishes the integral as a random variable. The result for the expectation applies to standard Brownian motion and the stated time interval; further details would be needed to present a complete proof or derive the variance independently.

Key ideas

• The expectation of the time integral of squared standard Brownian motion can be computed by interchanging integration and expectation.
• At time s, standard Brownian motion has second moment s.
• Integrating that second moment from zero to t yields an expectation of one half times t squared.
• The variance calculation is attributed to Itô’s isometry but is not shown in the document.

Tags

# integration of squared brownian motion w.r.t time


# integration of squared brownian motion w.r.t time












How to prove $\int_0^1 B_s^2ds$ is a random variable and compute its first two moments? From excercise 1.15 on the book martingales and brownian motion.

## Answer by Kevin (score 2, accepted)

https://quant.stackexchange.com/a/48981

The expectation follows from Fubini since $\mathbb{E}\left[\int_0^t B_s^2 \mathrm{d}s\right] = \int_0^t \mathbb{E}[B_s^2] \mathrm{d}s= \int_0^t s\mathrm{d}s = \frac{1}{2}t^2$.

The variance follows from Ito's Isometry and is answered here.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.