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11 Stochastic Processes

11 Stochastic Processes - 241B Lecture Stochastic Processes...

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Unformatted text preview: 241B Lecture Stochastic Processes Martingales Let X t be an element of Z t . Then X t is a martingale with respect to Z t if E [ X t j Z t & 1 ;Z t & 2 ;:::;Z 1 ] = X t & 1 for all t & 2 : ¡ The collection ( Z t & 1 ;Z t & 2 ;::: ) is called the information set at t ¢ 1 ¡ If the conditioning information set is ( X t & 1 ;X t & 2 ;::: ) , then X t is a martin- gale (it is implicit that X t is a martingale with respect to X ) ¡ If X t is a martingale with respect to Z t then X t is a martingale (because Z t contains X t ) ¡ The vector Z t is a martingale if E [ Z t j Z t & 1 ;Z t & 2 ;:::;Z 1 ] = Z t & 1 for all t & 2 ¡ If the process started in the in&nite past, there is no need to include the quali&er ¡for all t & 2 ¢ ¡ For our large-sample results, it does not matter if the process started in the in&nite past (simply that the process started before the &rst observation) Random Walks A leading example of a martingale process is a random walk. Let f U t g be vector independent white noise, so EU t = 0 and the covariance matrix of U t is &nite. A random walk is a sequence of cumulative sums Z 1 = U 1 ;Z 2 =...
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11 Stochastic Processes - 241B Lecture Stochastic Processes...

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