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Solutions to Problems to Work for Understanding

Adapted from Exercise 3.1 in Financial Calculus by M. Baxter and A. Rennie, Cambridge University Press, 1996, page 49. If Z is a normal N(0,1) random variable, the process is X(t) = \sqrt(t)*Z is distributed as a normal random variable at every time, X(0) = 0, and it is continuous at the origin t = 0. Is X(t) Brownian Motion?

The answer is no. The increment X(t+s) - X(s) = \sqrt(t+s)*Z - \sqrt(s)*Z = ( \sqrt(t+s) - \sqrt(s) )*Z is normal with mean zero, but variance ( (t+s) - 2\sqrt( s*(t+s) ) + s ) which is not t and the increment is not independent of X(s).

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