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Unformatted text preview: Chapter 13 The Strong Markov Property and Martingale Problems Section 13.1 introduces the strong Markov property — indepen- dence of the past and future conditional on the state at random (optional) times. Section 13.2 describes “the martingale problem for Markov pro- cesses”, explains why it would be nice to solve the martingale prob- lem, and how solutions are strong Markov processes. 13.1 The Strong Markov Property A process is Markovian, with respect to a filtration F , if for any fixed time t , the future of the process is independent of F t given X t . This is not necessarily the case for a random time τ , because there could be subtle linkages between the random time and the evolution of the process. If these can be ruled out, we have a strong Markov process. Definition 131 (Strongly Markovian at a Random Time) Let X be a Markov process with respect to a filtration F , with transition kernels μ t,s and evolution operators K t,s . Let τ be an F-optional time which is almost surely finite. Then X is strongly Markovian at τ when either of the two following (equivalent) conditions hold P ( X t + τ ∈ B |F τ ) = μ τ,τ + t ( X τ , B ) (13.1) E [ f ( X τ + t ) |F τ ] = ( K τ,τ + t f )( X τ ) (13.2) for all t ≥ , B ∈ X and bounded measurable functions f ....
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- Spring '06
- Markov process, Markov chain, Andrey Markov, Markov Property, Markovian, strong markov