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HO4-08 - Math 236 Stochastic Dierential Equations Problem...

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Math 236, Stochastic Differential Equations Problem set 4 February 17, 2009 These problems are due on Tuesday February 24 . 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 Let X ( t ) = ( X i ( t )) be the n -vector valued Ornstein-Uhlenbeck process. We use arguments for the time since we want to use subscripts for the components of the vector. Its SDE is dX ( t ) = AX ( t ) dt + σdB ( t ) , X (0) = x where A and σ are n × n matrices and B ( t ) = ( B i ( t )) is a vector of n independent standard Brownian motions. Show (i) that the covariance matrix C ( t ) = E x { X ( t ) X ( t ) T } satisfies the matrix differential equation (Lyapounov equation) dC ( t ) dt = AC ( t ) + C ( t ) A T + σσ T , C (0) = xx T Here the superscript T denotes transpose. Find (ii) an integral representation for the covariance matrix C ( t ). Show (iii) that if all the eigenvalues of A have negative real parts then the equilibrium (time homogeneous) OU process has the form X e ( t ) = integraldisplay t -∞ e A ( t - s ) σdB s
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