lecture-13 - Chapter 13 The Strong Markov Property and...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

Page1 / 3

lecture-13 - Chapter 13 The Strong Markov Property and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online