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HO2 - Math 236 Stochastic Dierential Equations Problem set...

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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 0 < t 1 < t 2 < · · · < t N = T be a partition of the interval [0 , T ] and let B t , t 0 , 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 0 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 | ≥ λ } ≤ E {| X | p } λ 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 σ 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 X t = x + integraldisplay t 0 b ( X s ) ds + integraldisplay t 0 σ ( X s ) dB s , assuming that it is well defined, with b ( x ), c ( x ) and σ ( x ) bounded and differentiable functions.
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