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HW7_F09

# HW7_F09 - Stats 219 Stochastic Processes Homework Set 7...

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Stats 219 - Stochastic Processes Homework Set 7, Fall 2009 1. Exercise 4.2.4 (a) This can be easily proved by replacing t + h with t and t with s in the identity as given in hints. Note that A t is a deterministic process and hence comes out of the conditioning. (b) Guassian ensures square integrable, independent increments come from the fact that Cov ( M t + h - M t , M t ) = 0, by martingale property. (c) This is readily seen from (a) and (b) 2. Exercise 4.2.5 . Let G t denote the canonical filtration of a Brownian motion W t . (a)Show that for any λ IR, the S.P. M t ( λ ) = exp( λW t - λ 2 t/ 2), is a continuous time martingale with respect to G t . (b) Explain why d k k M t ( λ ) are also martingales with respect to G t . (c) Compute the first three derivatives in λ of M t ( λ ) at λ = 0 and deduce that the S.P. W 2 t - t and W 3 t - 3 tW t are also MGs. ANS: (a) let Y t = e λW t - λ 2 ( t/ 2) . Then, E | Y t | = e - λ 2 ( t/ 2) E [ e λW t ] which since W t is a Gaussian ran- dom variable, we know to be finite. Also, E e λ ( W t + h - W t ) = e λ 2 h/ 2 yielding the identity E [ Y t + h |F t ] = e - λ 2 ( t/ 2)+ λW t = Y t so Y t is a martingale. (b) The filtration in question does not involve λ and differential operator and expectation operator is clearly exchangeable due to the nice nature of exponential function,i.e. d d λ M s ( λ ) = d d λ E [ M t ( λ ) |F s ] = E [ d d λ M t ( λ ) |F s ]. Hence d d λ M t ( λ ) is a martingale. Likewise, we repetitively apply differential operator to prove that all the derivatives are still martingales. (c) M 0 t (0) = exp( λW t - λ 2 t/ 2)( W t - ) | 0 = W t M 00 t (0) = exp( λW t - λ 2 t/ 2)( W t - ) 2 + exp( λW t - λ 2 t/ 2)( - t ) | 0 = W 2 t - t M 000 t (0) = exp( λW t - λ 2 t/ 2)( W t - ) 3 +exp( λW t - λ 2 t/ 2)( - 2)( W t - ) λ

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