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Orbit
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PWOQF, expected hitting time

May 16th, 2021, 6:59 pm

Hi all:in PWOQF Volume 1, p.161, 162 Paul says:

Let me introduce the function [$]C(y,t;t')[$] as the probability of the variable [$]y[$] leaving the region [$]\Omega[$] before time [$]t'[$]. This function can be thought of as a cumulative distribution function.

...Once we have found [$]C[$] then it is simple to find expected first exit time. Let me call the first exit time [$]u(y,t)[$].  Because [$]C[$] is a cumulative distribution function the expected first exit time can be written as
$$ u(y,t)= \int_{t}^{\infty}(t'-t)\frac{\partial C}{\partial t'}dt'.$$
After an integration by parts we get
$$ u(y,t)= \int_{t}^{\infty}1-C(y,t;t')dt'.$$
I'm sorry, maybe I'm losing it but I can't seem to make the integration by parts work. Can someone please advise?
 
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Cuchulainn
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Re: PWOQF, expected hitting time

May 16th, 2021, 7:16 pm

Hi,
you mean page 177?

I think the integral term should just have [$]C(y,t;t')[$]. What about eq. 10.5 after that?
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Orbit
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Re: PWOQF, expected hitting time

May 16th, 2021, 7:24 pm

Hi Cuchulainn, my version is 2003 I think?
Edit: in my version, equation 10.5 is the Kolomogorov equation, only instead of treating the probability function,it's treating the expected hitting time?
 
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Re: PWOQF, expected hitting time

May 16th, 2021, 7:51 pm

deleted -- later
 
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Alan
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Re: PWOQF, expected hitting time

May 16th, 2021, 8:17 pm

At first I thought it was ok, but now I think that you have to distinguish the case where the expected exit time is +infinity from the case where it's finite. In the latter case, by introducing the complementary density, you can show the claimed relation is correct. In the former case, the last integral just doesn't exist. I haven't looked at the text to see if the +infinity case is ruled out by the problem setup.
 
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Orbit
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Re: PWOQF, expected hitting time

May 16th, 2021, 8:24 pm

Thank you very much Alan.
1- Can you offer any pointers to finding what underlying parameters cause exit time to go unbounded vs. finite?
2- Can you guide me through (in the proper context) showing that the second integral is true?
 
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Alan
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Re: PWOQF, expected hitting time

May 16th, 2021, 8:54 pm

For 2, express the hitting time density as the negative t' derivative of a complementary distribution func. Since, by assumption in that case, the expected hitting time is finite, the hitting time density must decay at large t' like [$] 1/(t')^{2 + \epsilon}[$] or faster, where [$]\epsilon > 0[$]. Use that fact to convince yourself the (new) parts term is zero. Then, revert in the remaining integral from the complementary dist func. to the regular dist func.   

For 1, I looked at the text and it seems we are talking about 1D diffusions where [$]\Omega[$] is a two-sided spatial interval. I haven't really checked it, but would check drifting Brownian motion, with a positive drift, starting from the origin, and the interval [$]\Omega = (a,\infty)[$], where [$]a < 0[$].  Is the expected exit time finite? Is having one boundary at infinity ruled out?    

Even if having one boundary always at infinity is ruled out, suspect one could cook up some two-sided curving boundaries where the curving was such that the expected exit time was still +infinity.
 
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Orbit
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Re: PWOQF, expected hitting time

May 16th, 2021, 9:31 pm

Alan, thank you.
 
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Alan
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Re: PWOQF, expected hitting time

May 17th, 2021, 5:56 pm

You're welcome.

Looks like the expected hitting time of a std BM to a sloping line is always finite, as per here
I thought this might lead to a simple example where the expected exit time was +infinity, but it doesn't.
 
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Cuchulainn
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Re: PWOQF, expected hitting time

May 22nd, 2021, 12:30 pm

Of course, you find (probability of) hitting time by solving a pde and/or MC, yes?
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Re: PWOQF, expected hitting time

May 22nd, 2021, 12:52 pm

Of course, you find (probability of) hitting time by solving a pde and/or MC, yes?
Not necessarily. There are also martingale arguments: see the link in my last post.
 
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Orbit
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Re: PWOQF, expected hitting time

May 23rd, 2021, 9:55 pm

Alan/Cuch:
Honestly, with greatest respect to the PDE gurus here, these days I generally do anything in the world to avoid solving PDEs, and usually try martingale approaches. Just my 2 cents, ha ha.
BTW Alan your link is really great.
 
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Re: PWOQF, expected hitting time

May 23rd, 2021, 10:27 pm

Alan/Cuch:
Honestly, with greatest respect to the PDE gurus here, these days I generally do anything in the world to avoid solving PDEs, and usually try martingale approaches. Just my 2 cents, ha ha.
BTW Alan your link is really great.
I reached that point approximately 30 years ago.
 
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Re: PWOQF, expected hitting time

May 24th, 2021, 3:51 am

@Orbit: I'm guessing, because I haven't looked at this for decades, that on integrating the dC/dt' bit you can add any constant you like. (Although this never gets mentioned explicitly in textbooks.) I chose -1 to simplify behaviour at infinity. Etc. Is that all you wanted? I think Alan's answer is excellent, but since you specifically mentioned a concern about the integration by parts it might have been a little bit over the top.

@Alan: You are right, I never worry about pathological cases. I trust that somewhere along the way something would go wrong if the answer were infinite. This approach has never let me down, and has left me with time to spend more fruitfully on things that interest me.

@Cuch: This first-exit time problem cropped up in the early days of this forum. Someone tried to use all the heavyweight martingale machinery for what is a two-line problem when approached sensibly. They failed miserably. It was sufficiently embarrassing for them that I still use it to this day as an example of a sad, and I fear incurable, affliction I am going to call σύμβολορροια or symbolorrhoea. 
 
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Re: PWOQF, expected hitting time

May 24th, 2021, 10:27 am

Alan/Cuch:
Honestly, with greatest respect to the PDE gurus here, these days I generally do anything in the world to avoid solving PDEs, and usually try martingale approaches. Just my 2 cents, ha ha.
BTW Alan your link is really great.
BTW 'avoid' because 1) too messy or 2) there are better methods? Serious question.

No two languages are ever sufficiently similar to be considered as representing the same social reality. The worlds in which different societies live are distinct worlds, not merely the same world with different labels attached. Edward Sapir

The diversity of languages is not a diversity of signs and sounds but a diversity of views of the world.
Wilhelm von Humboldt

A good example is SABR SDE. One approach is to apply the Lamperti transform to get rid of the mixed derivatives (lots of campfire stories here) in the resulting horrendous PDE. Plan B is map the PDE to canonical form (see Courant Hilbert Vol II) i.e. no mixed derivative. Plan C is take the bull by the horns and solve untransformed PDE by FDM (but not ADI!).

I once showed the BS PDE to one of my old singular perturbation contacts. He started at the PDE for a while and finally uttered "where's the [$]\varepsilon[$]?".
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