Estimating Finite-Time Maxima and Minima of a Stationary Gaussian Ornstein-Uhlenbeck Process by Monte Carlo Simulation

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Abstract
In a stationary Ornstein-Uhlenbeck process, the Guassian variable y(0, 1) has serial correlation given by where ρδ t is the correlation between observations separated by time interval δ t , and the constant a is such that e-a is the correlation between observations separated by one unit of time. In any finite length of time, say m hours, there will be a minimum and a maximum of y whose probability distributions are of great operational importance, but which have never been determined with exactness. The author has approximated the distribution of min y(t) in time interval 0 <t < T by Monte Carlo simulation, T varying from one minute to one full month. While the resulting graph was prepared for hourly correlation ρ0 =0.95 it can be used for any other positive serial correlation by a shift of horizontal axis. By change of sign, it is usable to obtain the probability distribution of the m-hour maximum.

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