HO2-sol-08

HO2-sol-08 - Math 236 Stochastic Differential Equations...

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Unformatted text preview: Math 236, Stochastic Differential Equations Solutions for problem set 2 February 5, 2009 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 ) 2 = T } = 1 . Use the Borel-Cantelli lemma and the Chebychev inequality P {| X | ≥ λ } ≤ E {| X | p } λ p with p = 4. Solution We first note that we need to assume here that N × max ≤ k ≤ N- 1 ( t k +1 − t k ) is of order T as N → ∞ . This means that the partition is not very different from a uniform one in which all the intervals have the same length Δ = t k +1 − t k , with N Δ = T . We will assume for simplicity in the calculations that we have a uniform partition. Let X k +1 = ( B t k +1 − B t k ) 2 − ( t k +1 − t k ), for k = 0 , 1 ,...,N − 1. These are independent random variables with mean zero and variance 2( t k +1 − t k ) 2 = 2Δ 2 . For δ > 0 fixed, let A N = {| N- 1 summationdisplay k =0 ( B t k +1 − B t k ) 2 − T | > δ } = {| N- 1 summationdisplay k =0 X k +1 | > δ } We want to show that...
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HO2-sol-08 - Math 236 Stochastic Differential Equations...

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