Parameter estimation of Ornstein–Uhlenbeck and CIR processes

I would like to estimate Ornstein–Uhlenbeck process' parameters via Kalman filter.

My process is the following one:

$\text{d}x_{t}=\alpha(\theta-x_{t})\text{d}t+\sigma\text{d}W_{t}$

I'm interested in CIR process, too:

$\text{d}x_{t}=\alpha(\theta-x_{t})\text{d}t+\sigma x_{t}^{\beta}\text{d}W_{t}$

and my goal is to find the values of $\alpha$, $\theta$ and $\beta$ using Kalman filter over a state-space representation of the process.

How may I describe such a process in a form suitable to state-space representation and Kalman filter?

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I tried to edit this for you but it hasn't gone through. there is no such term $x_{t}^{\beta}$ in a classical OU process. what you have up there is more like a CIR process. you say don't mind the diffusion though, so whatever. – Veeken May 11 '13 at 19:39
CIR, you're right. – Lisa Ann May 11 '13 at 19:41
@Veeken I rejected your edit because changing an equation can fundamentally alter the question. If you suspect a formula is wrong, the best bet is to leave a comment as you've done here. All the same, thank you for pointing-out a suspicious post. – chrisaycock May 11 '13 at 21:17
@chrisaycock No problem with his correction, that was just my oversight to mix up UO with CIR formula. By the way, I decided to include both in my question :) – Lisa Ann May 11 '13 at 21:48
Well, you can find $\beta$ and $\sigma$ by using the quadratic variation of a process - no filtering is needed in such a case. W.r.t. $\alpha$ and $\theta$ you can use the UKF as Veeken suggested, or perhaps some particle filters. I am not an expert on filtering, but I'm pretty sure that both methods have their own advantages in your case. – Ilya May 21 '13 at 11:52

This book goes through exactly this problem in quite detail (with C++ codes included). I've worked through it in the past, but can't sum it up off the top of my head.

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this question can be quite straightforward or gnarly depending on whether you can observe measurements of $x_t$ directly or not. in the latter case, in general it will become a nonlinear system, and will require application of the extended kalman filter or its improvement, the unscented kalman filter.

On Edit: Now that the bounty has expired, let me answer by way of supplying this key reference (which i found as a result of this question*): "Estimating and Testing Exponential-Affine Term Structure Models by Kalman Filter" by Jin-Chuan Duan, Jean-Guy Simonato. Pg 13 of the paper gives the answer for the Vasicek model, page 15 for the CIR.

*Because of the results of Duffie and Kan (93), both these models lead to pricing equations for zero coupon bonds that are affine in the short rate, xt. As a result, one can use a linear Kalman filter to solve this, using zero coupon rate changes as the input. I originally thought one would definitely need the UKF.

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