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

Let W(t) be standard Brownian motion. What is Cov[W(3), W(7)]?

The covariance is 3. For standard Brownian motion the covariance between W(s) and W(t) equals the smaller of s and t, because the later value equals the earlier one plus an independent increment.

  1. A0
  2. B3Correct
  3. C7
  4. D10
  5. 21

Explanation

Write W(7) = W(3) + [W(7) − W(3)], with the increment independent of W(3). Cov = Var[W(3)] = 3. The value 7 uses the later time, and 21 multiplies the times.

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