HO1 - Math 236 Stochastic Differential Equations Problem...

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Unformatted text preview: Math 236, Stochastic Differential Equations Problem set 1 January 13, 2009 These problems are due on Tuesday January 20 . You can give them to me in class, drop them in my office, or put them in my mailbox outside the mathematics department. Problem 1 The stationary Ornstein-Uhlenbeck process Y ( t ) is a Gaussian process with mean zero and covariance R ( t,s ) = 1 2 e-| t- s | . Let N ( t ) be the standard Poisson process and define the process X ( t ) = ξ (- 1) N ( t ) , where ξ is a random variable independent of the Poisson process that takes values ± 1 with probability one half. Clearly X ( t ) takes only two values, ± 1. Show that it is stationary and that its covariance is e- 2 | t- s | . Thus the OU process Y ( t ) and Z ( t ) = 1 √ 2 X ( t 2 ) are both stationary and have the same covariance but are very different processes. Does X ( t ) satisfy the Kolmogorov condition for path continuity? Does the OU process? Show that these two processes are stochastically continuous.are stochastically continuous....
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This note was uploaded on 11/08/2009 for the course MATH 236 taught by Professor Papanicolaou during the Winter '08 term at Stanford.

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HO1 - Math 236 Stochastic Differential Equations Problem...

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