Timeline for On necessity of Feller property
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 26, 2017 at 17:44 | comment | added | Henry.L | @JohnDawkins Thanks, you seem to correct one of my long misconception! | |
| Jul 26, 2017 at 16:34 | comment | added | John Dawkins | q-l-c is foiled by a jump at a predictable stopping time. Here's a simple example of such. The state space is $[0,2]$. If $X_0\in[1,2]$ then $X$ moves to the right at unit speed until reaching $2$ where it is absorbed; that is, $X_t=X_0+t$ for $0\le t<2-X_0$ and $X_t=2$ for $t\ge 2-X_0$. If $X_0\in[0,1)$ then $X_t=X_0+t$ for $0\le t<1-X_0$ and $X_t=2$ for $t\ge 1-X_0$. This process has cadlag paths, but jumps at a predictable time if $X_0\in[0,1)$. | |
| Jul 25, 2017 at 21:52 | comment | added | Henry.L | @JohnDawkins Yes, can you construct an example that $X$ is defined on a compact space $E$ and $X$ is cadlag but not quasi-left-continuous. | |
| Jul 25, 2017 at 21:38 | comment | added | John Dawkins | @Henry.L: Can you be more specific about the sort of counter-example that would interest you? | |
| Jul 25, 2017 at 17:26 | comment | added | Henry.L | @JohnDawkins I do apologize for (2) again. | |
| Jul 25, 2017 at 17:25 | comment | added | Henry.L | (1) I mean when $X$ is defined on a compact space $E$, if you can find a counter example I would be interested to know. When $E$ is not compact sure. (2) Yes. I said a Hunt process is a F-D if if must possess F-D property then there is a one-to-one correspondence in my last comment. | |
| Jul 25, 2017 at 13:43 | comment | added | John Dawkins | @Henry.L: (2) I can only repeat that a Hunt process need not be a Feller-Dynkin process. I can supply an example if you wish. | |
| Jul 25, 2017 at 13:43 | comment | added | John Dawkins | @Henry.L: (1) A Hunt process is cadlag, but is also quasi-left continuous (meaning that if $(T_n)$ is an increasing sequence of stopping times with limit $T$, then $\lim_nX_{T_n}=X_T$ on the event $\{T_n<T,\,\forall n\}$.) For example, if $B$ is a standard one-dimensional reflecting Brownian motion (on $[0,\infty)$) and if $T_1:=\inf\{t>0:B_t=1\}$, define $X_t=B_t$ for $0\le t<T_1$ but $X_t=-1$ for $t\ge T_1$. Then $X$ is a cadlag strong Markov process (with state space $[0,1)\cup(1,\infty)\cup\{-1\}$) but $X$ is not a Hunt process, because q-l-c fails at $T_1$. | |
| Jul 24, 2017 at 22:37 | comment | added | Henry.L | @JohnDawkins (1) $X$ is compact so what is the difference between these two notions? I think they are the same. One step further if Hunt proc is also feller process then just take its cadlag version. (2) Hunt process must also posses Feller-Dynkin property as in OP. | |
| Jul 24, 2017 at 17:36 | comment | added | John Dawkins | @Henry.L: Two points: (1) the sample paths of a Hunt process are not merely "left-limited" but more stringently "quasi-left-continuous"; (2) the transition semigroup of a Hunt process is not necessarily Feller-Dynkin. | |
| Jul 24, 2017 at 15:50 | vote | accept | sharpe | ||
| Jul 24, 2017 at 15:50 | |||||
| Jul 24, 2017 at 13:03 | history | edited | Henry.L | CC BY-SA 3.0 |
added 87 characters in body
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| Jul 21, 2017 at 18:33 | history | answered | Henry.L | CC BY-SA 3.0 |