Timeline for Is there a theory of SDEs whose coefficients are themselves adapted processes (i.e. "may depend on the past")?
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| Mar 23, 2015 at 12:18 | answer | added | imateapot | timeline score: 1 | |
| Sep 12, 2013 at 17:41 | comment | added | David | @Joris: Thank you, that is an interesting suggestion. But lets say $W_t$ is Brownian motion on $\mathbb{R}$ and $\mu$ is a drift of strength $-1/(1-x)$ until $W_t$ hits -1, and then it is zero. (The first drift is the h-transform when you condition Brownian motion to hit $-1$ before $1$, so $W_t$ should be Brownian under this conditioning). Then Novikov's criterion will not be satisfied, since standard Brownian motion will go close to 1, where the drift blows up, with positive probability. So I cannot use Girsanov, right? | |
| Sep 12, 2013 at 9:07 | comment | added | Joris Bierkens | @ David: Did you consider the implications of Girsanov theorem for this setting? Let $\alpha_t = \mu((W_s)_{0 \leq s \leq t}$. By a change of measure of the form $\exp\left(\int_0^t \alpha_s d W_s - 1/2 \int_0^t \alpha_s^2 \ ds \right)$ (perhaps with different sign before the Ito integral) your process $W$ transforms into a Brownian motion. Girsanov theorem is classical and can be found everywhere e.g. in Oksendal or Karatzas/Shreve. | |
| Sep 11, 2013 at 14:10 | history | edited | Ricardo Andrade |
edited tags
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| Sep 11, 2013 at 14:05 | history | edited | Ricardo Andrade |
created tag 'stochastic-diff-equations'; added top level tag
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| Sep 11, 2013 at 13:57 | comment | added | The Bridge | @ David : You should have a look at Functional Itô calculus (papers.ssrn.com/sol3/papers.cfm?abstract_id=1435551 or proba.jussieu.fr/pageperso/ramacont/papers/…) Regards | |
| Sep 11, 2013 at 13:09 | history | edited | David | CC BY-SA 3.0 |
added 8 characters in body
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| Sep 11, 2013 at 12:54 | review | First posts | |||
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| Sep 11, 2013 at 12:39 | history | asked | David | CC BY-SA 3.0 |