HO2 - Math 236, Stochastic Differential Equations Problem...

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Unformatted text preview: Math 236, Stochastic Differential Equations Problem set 2 January 22, 2009 These problems are due on Friday January 30. 5pm . 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 0 = t < t 1 < t 2 < < t N = T be a partition of the interval [0 ,T ] and let B t , t , be the standard Brownian motion process. Show that N- 1 summationdisplay k =0 ( B t k +1- B t k ) 2 converges with probability one to T as N and max k N- 1 ( t k +1- t k ) 0. That is P { lim N N- 1 summationdisplay k =0 ( B t k +1- B t k ) = T } = 1 . Use the Borell-Cantelli lemma and the Chebychev inequality P {| X | } E {| X | p } p with p = 3. Problem 2 Let u ( t,x ) be a smooth (classical) solution of the terminal value PDE u t ( t,x ) + 1 2 2 ( x ) u xx ( t,x ) + b ( x ) u x ( t,x ) + c ( x ) u ( t,x ) = 0 , t < T , x R with terminal condition u ( T,x ) = f ( x ). Let X t be the Ito process defined by...
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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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HO2 - Math 236, Stochastic Differential Equations Problem...

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