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HO3-08

# HO3-08 - Math 236 Stochastic Dierential Equations Problem...

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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 + σ ( X t ) dB t , X 0 = x where b ( x ) and σ ( x ) satisfy the Ito conditions (linear growth bound and Lipschitz condition). Let x ( α, β ) and let τ = τ x = inf { t > 0 : X t [ α, β ] c } be the exit time from the interval [ α, β ] starting from x in its interior, and let L be the generator of the process L = 1 2 σ 2 ( x ) 2 ∂x 2 + b ( x ) ∂x Show using Ito’s formula and the optional stopping theorem that if there is a non-negative and bounded function u ( x ) for x [ α, β ] such that L u ( x ) ≤ - 1, then E x { τ } < and so τ x < with probability one. Problem 2 With the notation of Problem 1, show that if the solution of the boundary value problem L u ( x ) = - 1 for x ( α, β ) with u ( α ) = u ( β

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HO3-08 - Math 236 Stochastic Dierential Equations Problem...

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