HO3-08 - Math 236, Stochastic Differential Equations...

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Unformatted text preview: Math 236, Stochastic Differential Equations Problem set 3 February 5, 2009 These problems are due on Friday February 13 . 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 be the one-dimensional diffusion process satisfying the Ito SDE dX t = b ( X t ) dt + ( X t ) dB t , X = x where b ( x ) and ( x ) satisfy the Ito conditions (linear growth bound and Lipschitz condition). Let x ( , ) and let = x = inf { t > 0 : X t [ , ] c } be the exit time from the interval [ , ] starting from x in its interior, and let L be the generator of the process L = 1 2 2 ( x ) 2 x 2 + b ( x ) x Show using Itos formula and the optional stopping theorem that if there is a non-negative and bounded function u ( x ) for x [ , ] such that L u ( x ) - 1, then E x { } < and so x < with probability one....
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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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HO3-08 - Math 236, Stochastic Differential Equations...

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