Timeline for Canonical Markov process and abstract Markov process
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 27, 2021 at 20:17 | comment | added | Oli Bernet | I see, I thought there might be some bigger advantage than just an ease for defining things, but that clears thing up thank you very much Mateusz | |
| Sep 27, 2021 at 19:43 | comment | added | Mateusz Kwaśnicki | First of all, the canonical probability space is not really canonical, as it can refer to any of the following: the class of all paths, the class of càdlàg paths, or the class of continuous paths. It is convenient to work with the canonical realisation of a Markov process (that is, the one defined on the canonical probability space): shift operators, time-reversal and other path transformations are then defined with no difficulties. On the other hand, it is often necessary to work in a general setting — in order to have more then one adapted process or variables independent of the process. | |
| Sep 27, 2021 at 16:05 | comment | added | Oli Bernet | It is actually similar to a Brownian motion. If a Brownian motion is given we can always find its canonical version, same process as above, but usually we just work with the abstract version, so what is the advantage of having a canonical version? | |
| Sep 27, 2021 at 16:00 | comment | added | Oli Bernet | Sure ,let a probability space be given and a Markov process $X$ on it (using the filtration generated by X) with state space $(E,\mathcal{E}).$ Then we can consider $X$ as a random map $X: \Omega \rightarrow , \omega \mapsto (t \mapsto X_t(\omega)) \in E^{[0,\infty)} $ where $E^{[0,\infty)} := \{f:[ 0,\infty) \mapsto E \}.$ Further, let us define the following process $Y$ on $E^{[0,\infty)}:$ \begin{align*} Y_t: &E^{[0,\infty)} \rightarrow E \\ &f \mapsto Y_t(f)= f(t) \end{align*} Then $Y$ is a Markov on $E^{[0,\infty)}$ in regard to distribution of $X.$ $Y$ is the canonical version | |
| Sep 27, 2021 at 15:01 | review | Close votes | |||
| Oct 13, 2021 at 1:44 | |||||
| Sep 27, 2021 at 14:43 | comment | added | Kostya_I | Could you add definitions or references? What's a canonical Markov process? | |
| Sep 27, 2021 at 14:16 | history | asked | Oli Bernet | CC BY-SA 4.0 |