SERVING THE QUANTITATIVE FINANCE COMMUNITY

Alan
Topic Author
Posts: 10264
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Exact sde solution for the sqrt model

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.

EBal
Posts: 431
Joined: May 20th, 2005, 1:30 pm

### Exact sde solution for the sqrt model

Substitution makes itthen the solution is trivial.

Alan
Topic Author
Posts: 10264
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Exact sde solution for the sqrt model

Yes -- that's it.

frenchX
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### Exact sde solution for the sqrt model

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.

Alan
Topic Author
Posts: 10264
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Exact sde solution for the sqrt model

I'll send you a copy.

frenchX
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### Exact sde solution for the sqrt model

Thank you very much Alan. Very kind of you, as usual.

Alan
Topic Author
Posts: 10264
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Exact sde solution for the sqrt model

Note they make a mistake in applying their method to this case.

secret2
Posts: 304
Joined: July 28th, 2010, 10:29 pm

### Exact sde solution for the sqrt model

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

Alan
Topic Author
Posts: 10264
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Exact sde solution for the sqrt model

Sure -- send me an email to the address in my profile.

croot
Posts: 104
Joined: July 23rd, 2006, 8:30 pm

### Exact sde solution for the sqrt model

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.

Alan
Topic Author
Posts: 10264
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Exact sde solution for the sqrt model

Well, to all those who are disappointed -- please find some additional exact solutions to spice things up!

list
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

### Exact sde solution for the sqrt model

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.

secret2
Posts: 304
Joined: July 28th, 2010, 10:29 pm

### Exact sde solution for the sqrt model

sqrt() is DEFINED as the positive root. Merry X'mas.

croot
Posts: 104
Joined: July 23rd, 2006, 8:30 pm

### Exact sde solution for the sqrt model

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?

Alan
Topic Author
Posts: 10264
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Exact sde solution for the sqrt model

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.

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...

 JOBS BOARD

Looking for a quant job, risk, algo trading,...? Browse jobs here...

GZIP: On