Editing Markov property (section)

==Strong Markov property==
Suppose that <math>X=(X_t:t\geq 0)</math> is a [[stochastic process]] on a [[probability space]] <math>(\Omega,\mathcal{F},P)</math> with [[natural filtration]] <math>\{\mathcal{F}_t\}_{t\geq 0}</math>. Then for any [[stopping time]] <math> \tau </math> on <math> \Omega </math>, we can define

:<math>\mathcal{F}_{\tau}=\{A \in \mathcal{F}:\forall t \geq 0, \{\tau\geq t\} \cap A \in \mathcal{F}_{t}\}</math>.

Then <math>X</math> is said to have the strong Markov property if, for each [[stopping time]] <math>\tau</math>, conditional on the event <math>\{\tau < \infty\}</math>, we have that for each <math>t\ge 0</math>, <math>X_{\tau + t}</math> is independent of <math>\mathcal{F}_{\tau}</math> given <math>X_\tau</math>.

The strong Markov property implies the ordinary Markov property since by taking the stopping time <math>\tau=t</math>, the ordinary Markov property can be deduced.<ref>Ethier, Stewart N. and [[Thomas G. Kurtz|Kurtz, Thomas G.]] ''Markov Processes: Characterization and Convergence''. Wiley Series in Probability and Mathematical Statistics, 1986, p. 158.</ref>
<!-- Hide for now ... name '''Markov-type property''' seems doubtful
==Markov Type Property==
A stochastic process has a '''Markov-type property''' if the process's [[random variable]]s determine a set of probabilities can be factored in a way that yields the '''Markov property'''. Useful in applied research, members of such classes{{Clarify|October 2009|date=October 2009}} defined by their mathematics or area of application{{Clarify|October 2009|date=October 2009}} are referred to as '''[[Markov random field]]s'''., and occur in many situations. The [[Ising model]] is a prototypical example. -->