Timeline for When is a continuous path stochastic process be representable as diffusion or Ito process?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 17, 2013 at 19:43 | comment | added | Hans | Regarding Martin's solution on the variance problem, could you please post your similar but different generalization? I am eager to see that. | |
| Dec 17, 2013 at 19:37 | comment | added | Hans | I am a little surprised that you feel "it's getting strange", because is my original question not asking for an example of continuous path (Markovian) STOCHASTIC process that is NOT a (Ito) diffusion? I like your new example $f(W_t)$ though. | |
| Dec 17, 2013 at 0:18 | comment | added | Yuri Bakhtin | You want a "stochastic" example of a process that is not a "diffusion"? Don't you think it's getting strange? Well, I don't know if this is gonna satisfy you: take that increasing non-absolutely continuous deterministic trajectory $f(t)$ and instead of time substitute a Wiener process obtaining $f(W(t))$. As for Martin's comment, I thought I understood what he was saying, especially because I had a similar generalization in mind (not quite the same), but why don't you ask him. Anyway, I have not thought much about the nonsymmetric case. | |
| Dec 16, 2013 at 20:02 | comment | added | Hans | By the way, do you understand Martin Hairer's strong Markov derivation, specifically about the law of $X(t)$ and $Y(t)$ being equal, in the 1-d continuous path Markov process increasing variance problem? Can you explain it to me there if you don't mind? | |
| Dec 16, 2013 at 18:38 | comment | added | Hans | Thank you for providing these examples and going along with me, Yuri. As you said, I don't like the example $x(t)=t^\alpha$ with $\alpha$, since I can still call it ODE and thus SDE. Your Update example is better. Still it is not a strictly stochastic example which I seek and would prefer. | |
| Dec 15, 2013 at 17:56 | history | edited | Yuri Bakhtin | CC BY-SA 3.0 |
added 138 characters in body
|
| Dec 15, 2013 at 17:19 | comment | added | Yuri Bakhtin | Right, i should have taken care of the infinite diffusion coefficient at zero. To that end, instead of constant speed motion, take $x(t)=t^a$ with $a$ small. You still can say that away from $0$ these trajectories solve an ODE and in the integral sense even including $0$. And here we again need a definition of a diffusion process. If it includes linear behavior of variance of the increment then this is not a diffusion. Of course, a time change also cures this singularity, so, unfortunately, this question is partially about definitions. | |
| Dec 15, 2013 at 13:38 | comment | added | Hans | Thank you, Yuri, for the explanation and reference. The counter example however does not seem to be correct. I suppose you want the two trajectory to be deterministic except at time 0? But ODE is just a special case of SDE, or $dx = \mbox{sign}(x)dt$, then assign velocity -1 and 1 to $x=0$ with equal probability. Why can't this be classified as diffusion? Regarding classification of diffusion, I suppose you are saying the Ito process is but one type of diffusions, the general definition of which, as you say, varies. Also, Dan has provided a reference for one sufficient and necessary condition. | |
| Dec 14, 2013 at 22:52 | history | answered | Yuri Bakhtin | CC BY-SA 3.0 |