# Exit time of interval by diffusion using Girsanov's theorem

Suppose we have a process $X$ with $dX_t=\sigma dB_t + \mu dt$, for constants $\sigma$ and $\mu$, started at $x\in (a,b)$, for some constants $a$ and $b$, where $B$ is a Brownian motion. We'd like to determine the probability that $X$ exits $[a,b]$ via $a$. This can be done by finding the scale function (as explained here, for example).

Is it also possible to solve this problem by applying Girsanov's Theorem? For similar problems, this approach tends to be pretty effective, as it let's us reformulate the problem as one involving hitting times of a Brownian motion, rather than a diffusion.

Thanks.

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@ Ben Derrett : I think it's possible, Girsanov would remove the drift and then you should be able to reduce the problem to brownian motion case using equality in law of $\sigma B_t$ with $B_{\sigma^2.t}$. Regards –  TheBridge Nov 22 '12 at 14:28
@TheBridge Well if you can produce a proof with Girsanov's Theorem, I'd be very happy. –  Ben Derrett Nov 22 '12 at 15:08
@ Ben Derrett : I think I was a little optimistic, what I have read is about applying Girsanov to calculate more easily Laplace transform of hitting time of a single barrier (for drift BM or GBM). Best regards –  TheBridge Nov 23 '12 at 7:19