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IAI Actuarial Core Principles · Economic Modelling · Stochastic models for security prices

For a standard Brownian motion W(t), what is Var[W(9) − W(4)]?

The variance is 5. Brownian motion has independent stationary increments with variance equal to the length of the time interval, and the interval from time 4 to time 9 has length 5.

  1. A3
  2. B5Correct
  3. C9
  4. D13
  5. 25

Explanation

Increments of standard Brownian motion over an interval of length h have a N(0,h) distribution. Here h = 9 − 4 = 5, so the variance is 5. The value 13 wrongly adds the times, and 9 uses only the end time.

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