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

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Math 236, Stochastic Differential Equations Solutions to problem set 3 February 17, 2009 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. Solution First we note that t τ are stopping times that are bounded since t τ t . From Ito’s formula we have du ( X s ) = L u ( X s ) + σ ( X s ) u ( X s ) dB s and integrating u ( X t τ ) = u ( x ) + integraldisplay t τ 0 L u ( X s ) ds + integraldisplay t τ 0 σ ( X s ) u x ( X s ) dB s Assuming that u x is bounded (not explicitly stated in the statement), taking expectations and using the optional stopping theorem we have E x { u ( X t τ ) } = u ( x ) + E x { integraldisplay t τ 0 L u ( X s ) ds } Using the hypotheses on u we have therefore that E x { t τ } ≤ u ( x ) Since t τ τ
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