Interview Questions VII: Integrated Brownian Motion

For a standard Weiner process denoted , calculate

This integral is the limit of the sum of at each infinitessimal time slice from to , it is called Integrated Brownian Motion.

We see immediately that this will be a random variable with expectation zero, so the result of the squared expectation will simply be the variance of the random variable, which is what we need to calculate. Here I show two ways to solve this, first by taking the limit of the sum and second using stochastic integration by parts

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1) Limit of a sum

(1)

This sum of values along the Wenier process is not independent of one-another, since only the increments are independent. However, we can re-write them in terms of sums of these independent increments

(2)

where are the individual independent increments of the brownian motion. Substituting into our previous equation and reversing the order of the summation

(3)

which is simply a weighted sum of independent gaussians. To calculate the total variance, we sum the individual variances using the summation formula for

(4)

which is the solution.

2) Stochastic integration by parts

The stochastic version of integration by parts for potentially stochastic variables , looks like this:

Re-arranging this gives

Now setting and we have

(5)

We recognise this as a weighted sum of independent gaussian increments, which is (as expected) a gaussian variable with expectation 0 and variance that we can calculate with the Ito isometry

(6)

which is the solution.