hmw1 - X ( t ) = x + e-at ( x-x ) + b Z t e-a ( t-s ) dW (...

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Numerical Methods for SDEs, Fall 2006. Course Instructor: Ra´ul Tempone. Homework Set 1, due Thursday Sept 7. Last revised, Aug 26, 2006. Exercise 1 Prove the following identities by taking limits of the Forward Euler method: Z T 0 tdW ( t ) = T W ( T ) - Z T 0 W ( t ) dt (integration by parts) Z T 0 W ( t ) dW ( t ) = W ( T ) 2 2 - T 2 Exercise 2 The Ornstein-Uhlenbeck process can defined by
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Unformatted text preview: X ( t ) = x + e-at ( x-x ) + b Z t e-a ( t-s ) dW ( s ) , where a, b > are given constants. Compute the expected value and the variance of X ( t ) . Then compute their limits as t . Interpret the results. 1...
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