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Solutions to Problems to Work for Understanding

Adapted from Exercise 3.2 in Financial Calculus by M. Baxter and A. Rennie, Cambridge University Press, 1996, page 49. If X_1(t) and X_2(t) are two independent Brownian motions and \rho is a constant, -1 < \rho < 1, then the process X(t) = \rho*X_1(t) + \sqrt( 1 - \rho^2)*X_2(t) is continuous at the origin, and has marginal distributions N(0,t). Is this X a Brownian motion?

Yes. To check this, it is necessary to verify the 4 components of the definition:

  1. The increment X(t+s) - X(s) is the sum of independent N(0,\rho^2*t) and N(0, (1-\rho^2)*t) random variables, so it is an N(0,t) random variable.
  2. The increment X(t_4) - X(t_3) is independent of X_1(t_2) - X(t_1) and also of X_2(t_2) - X_1(t_1) and so is independent of their weighted sum.
  3. X(0) = \rho*X_1(0) + \sqrt( 1 - \rho^2 )*X_2(0) = 0
  4. lim_{t \to 0} X(t) = \rho*lim_{t \to 0} X_1(t) + \sqrt( 1 - \rho^2 )*lim_{t \to 0} X_2(t) = 0

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