Markov_note

Markov_note - Stat/219 Math 136 Stochastic Processes Notes...

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Notes on Markov Processes 1 Notes on Markov processes The following notes expand on Proposition 6.1.17, and attempt to minimize confusion between the terms “stationary” and “homogeneous”. 1.1 Stochastic processes with independent increments Lemma 1.1 If { X t ,t 0 } is a stochastic process with independent increments then { X t ,t 0 } is a Markov process. Proof: Let {F t ,t 0 } be the canonical Fltration of { X t ,t 0 } . Independent increments means that for any t,h 0, the random variable X t + h X t is indenpendent of F t . To show that { X t } is a Markov process, from DeFnition 6.1.10 it su±ces to show that IE[ f ( X t + h ) |F t ] = IE[ f ( X t + h ) | X t ] for any bounded measurable function f on ( S , B ) (where S is the state space and B is its Borel σ -Feld). Let f be such a function. Then IE[ f ( X t + h ) |F t ] = IE[ f ( X t + h X t + X t ) |F t ] = IE[ f ( X t + h X t + X t ) | X t ] , where the last equality follows upon noting that X t is F t -measurable and X t + h X t is independent of F t . (This last statement is a consequence of the “independence” lemma for conditional expectations, Lemma 1.3 below). The converse of Lemma 1.1 is not true. Try to Fnd a counterexample. (Hint: consider a process from Exercise 6.1.19.) 1.2 Homogeneous Markov processes The main result is Proposition 6.1.17: If { X t ,t 0 } has stationary and independent increments then { X t ,t 0 } is a homogeneous Markov process. The idea of the proof is similar to the proof of Lemma 1.1 above.

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Markov_note - Stat/219 Math 136 Stochastic Processes Notes...

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