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Alan
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Exact sde solution for the sqrt model

December 22nd, 2011, 7:34 pm

From a problem raised in the student forum:With sigma > 0 fixed, the celebrated sqrt model SDE (below) has a closed-form SDE solution (for all time) for at least one value of omega = omega(sigma). What is this special omega and what is the solution?
Last edited by Alan on December 22nd, 2011, 11:00 pm, edited 1 time in total.
 
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EBal
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Exact sde solution for the sqrt model

December 23rd, 2011, 12:52 pm

Substitution makes itthen the solution is trivial.
 
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Alan
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Exact sde solution for the sqrt model

December 23rd, 2011, 2:21 pm

Yes -- that's it.
 
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frenchX
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Exact sde solution for the sqrt model

December 23rd, 2011, 2:22 pm

And that's the unique case for which you have a closed form solution ? That's a bit disappointing ... I'd be very interested in reading the paper you quoted Alan about solution of 1D SDE. "Solutions and Simulations of Some One-Dimensional Stochastic Differential Equations "QuoteWe consider a one dimensional SDE dX t = μ(X t )dt + σ(X t )dB t . We give a new general formula for solutions that involves solving an associated ordinary differential equation. Explicit solutions are obtained in cases where the ODE has such. I'm really curious to see what is the corresponding ODE.
Last edited by frenchX on December 22nd, 2011, 11:00 pm, edited 1 time in total.
 
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Alan
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Exact sde solution for the sqrt model

December 23rd, 2011, 2:29 pm

I'll send you a copy.
 
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frenchX
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Exact sde solution for the sqrt model

December 23rd, 2011, 2:30 pm

Thank you very much Alan. Very kind of you, as usual.
 
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Alan
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Exact sde solution for the sqrt model

December 23rd, 2011, 2:51 pm

Note they make a mistake in applying their method to this case.
 
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secret2
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Exact sde solution for the sqrt model

December 23rd, 2011, 3:48 pm

If that's not too much trouble Alan I'd also love to have a copy. You know, it's X'mas long weekend and it's good to have some entertainment
 
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Alan
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Exact sde solution for the sqrt model

December 23rd, 2011, 4:22 pm

Sure -- send me an email to the address in my profile.
 
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croot
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Exact sde solution for the sqrt model

December 24th, 2011, 8:49 am

I'm disappointed: at least a countable family corresponding to the integral dimensions of a related BESQ process !
Last edited by croot on December 23rd, 2011, 11:00 pm, edited 1 time in total.
 
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Alan
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Exact sde solution for the sqrt model

December 24th, 2011, 3:32 pm

Well, to all those who are disappointed -- please find some additional exact solutions to spice things up!
 
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list
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Exact sde solution for the sqrt model

December 24th, 2011, 6:30 pm

QuoteOriginally posted by: EBalSubstitution makes itthen the solution is trivial.It looks like Y takes values of two signs though sqrt ( X ) usually is interpreted as a positive one.
Last edited by list on December 23rd, 2011, 11:00 pm, edited 1 time in total.
 
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secret2
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Exact sde solution for the sqrt model

December 24th, 2011, 7:07 pm

sqrt() is DEFINED as the positive root. Merry X'mas.
 
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croot
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Exact sde solution for the sqrt model

December 25th, 2011, 12:22 pm

Are we agreed that integer multiples of Ebal's answer give 'a closed-form SDE solution (for all time)'?The argument seems to be that a Bessel of integer dimension is 'a closed-form SDE solution'..It would be interesting to better define this notion. After all, what makes an OU more closed form than a Bessel?
 
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Alan
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Exact sde solution for the sqrt model

December 25th, 2011, 7:07 pm

Not agreed at all. I didn't insist that Ebal post his 'trivial' solution, but it follows from the well-known OU solution as given, say at wikipedia:This is closed-form because, given the driving Brownian motion B(t), it explicitly contructs "a solution" X(t) as a functional of time and B(t).With that definition of closed-form, nobody has shown any other one, so far. (Technically, there are an infinity of solutionssince the origin is regular here, but they all agree with the one shown up to the first hitting time of 0)Now, it's true that, for integer d=1,2,3,... you could drive d independent OU processes by running d independent Brownian motionsin a d-dimensional space. The law of the square of the radial distance from the origin of that process is indeed the same as the law of X(t) -- for special omegas. In other words, it's true that Feller's Sqrt process isequivalent to a Squared Radial OU process. But this does not yield a closed-form soln to the original problem under my definition. If somebody would like to argue for why a different defn is better or whyI'm out to lunch, please go ahead.
Last edited by Alan on December 24th, 2011, 11:00 pm, edited 1 time in total.
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