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Is there an existence and uniqueness theorem for SDEs of the following type:

$dW_{t}=d\tilde{W}_{t}+\mu\left(\left(W_{s}\right)_{0\le s\le t},t\right)dt$,

where $\tilde{W}_{t}$ is say $d$-dimensional Brownian motion and where $\mu$ is a "nice" function of the path of the solution up to time t. (In the case I am interested in, it's a drift that turns on or off depending on whether certain $W_t$-stopping times have happened by time $t$ or not).

I could definitely construct the solution by "gluing together" solutions for intervals of time during which the drift only depends on the current position $W_t$, but I was wondering if there is a slicker way.

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  • $\begingroup$ @ 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 $\endgroup$
    – The Bridge
    Commented Sep 11, 2013 at 13:57
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    $\begingroup$ @ 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. $\endgroup$
    – Joris Bierkens
    Commented Sep 12, 2013 at 9:07
  • $\begingroup$ @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? $\endgroup$
    – David
    Commented Sep 12, 2013 at 17:41

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Existence and uniqueness theorems under usual regularity conditions can be found, for instance, in book 1 of Liptser & Shyriaev "Statistics of Random Processes".

However, because your drift switches on and off, gluing together pieces between stopping times seems to me the slickest way.

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