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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:

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.

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.

X(0) = \rho*X_1(0) + \sqrt( 1  \rho^2 )*X_2(0) = 0

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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