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- mixmasterdeik
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First Passage time
Hi,if I have a brownian motion, what's the expected first passage time for it to cross a level m? The application I am trying to do is for a rate process following a normal or lognormal distribution currently at 10%, what;'s the expected time for it to cross X bps from 10% given a normal vol or lognormal vol, depending on the process assumption.Thks
First Passage time
Hi, If I remember correctly the first passage time of a BM has infinite expectation.
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- DevonFangs
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First Passage time
You have to use reflection principle and you'll have the probability distribution of the first passage time.
First Passage time
For a standard BM starting at 0, I think the expected first time to touch -a or b is the product a.bSo if you have just one barrier, the expectation is infinite.
First Passage time
in the book The theory of stochastic processes III, Springer at the and of the book they studied one dimensional SDEs. There you can find some explicit formulas related to f.p.t
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- Cuchulainn
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First Passage time
You can find first exit time by solving a benign PDE (Dirichlet BC, no truncation stuff) or using MC simulation. Is OP question 1 factor?
Last edited by Cuchulainn on January 6th, 2012, 11:00 pm, edited 1 time in total.
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- Cuchulainn
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First Passage time
So, it looks like it is a question of finding/computing the density function for a random process to cross a given barrier?For the rates example, if you use eq. 1.1 (Patie article) and then 1.5 which can then be easily solved. I have some code for this; if you a data set I can test it.
Last edited by Cuchulainn on January 12th, 2012, 11:00 pm, edited 1 time in total.
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- mixmasterdeik
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First Passage time
Hi, I am looking at eq 3.7.3 of Shreve's Stochastic for Finance II book, which seems to have the density of the first passage time of a Brownian motion.On my original problem, I have a rate following a Brownian Motion process scaled by its normal volatility (arithmetic brownian motion with zero drift). Let's say starting point of the rate process is 10% and normal vol is sigma. Let's say I want to compute the first passage time of that variable to either 9.5% or 10.5%.Do u have a closed form formula for this expected first passage time? Do u have it for a geometric brownian motion also?Many thks
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- Cuchulainn
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First Passage time
It would seem that a closed solution is discussed herehttp://eprints.maths.ox.ac.uk/1383/1/T_Primozic.pdf (eq. (3)?)For more complicated processes closed solution will be more elusive and then we will need PDE or MC I reckon.
Last edited by Cuchulainn on January 12th, 2012, 11:00 pm, edited 1 time in total.
First Passage time
One can also obtain some FP distribution with a Laplace transform. If you have the probability density then multiply by t and integrate, you would have the expected first passage time (and you will find the result of ZhuLiAn, id est infinite expectation).Have a look here:http://www.puc-rio.br/marco.ind/hittingt.html
Last edited by frenchX on January 12th, 2012, 11:00 pm, edited 1 time in total.
First Passage time
There is this paper i read on the topic http://www.actuaries.org/AFIR/Colloquia ... n_Yor.pdfI got bored half way through and it was a while back so i dont remember much. Sounds similar to what frenchX is saying.