IEOR 4701: Stochastic Models in Financial Engineering Summer 2007, Professor Whitt SOLUTIONS to Homework Assignment 9: Brownian motion

The probability law of a stochastic process is usually specified by giving all the finitedimensional distributions (f.d.d.’s). Let {X(t) : t ≥ 0} be a stochastic process; i.e., a collection of random variables indexed by the parameter t, which is usually thought of as time. Then the f.d.d.’s of this stochastic process are the collection of k-dimensional probability distributions of the random vectors (X(t1), . . . , X(tk)), over all possible positive integers k and all possible vectors (t1, . . . , tk) and (x1, . . . , xk) with 0 < t1 < · · · < tk. The multivariate cdf of X(t1), . . . , X(tk) is given by