Estimating Ornstein–Uhlenbeck and CIR Processes with Kalman Filters
Summary
The document asks how to estimate the mean-reversion parameters of Ornstein–Uhlenbeck and Cox–Ingersoll–Ross style processes from a state-space model. It distinguishes the constant-volatility OU process from a process whose volatility depends on the state, and raises the challenge of estimating parameters such as mean-reversion speed, long-run level, and volatility exponent with a Kalman filter.
The responses explain that the right filter depends on what is observed. If the process is observed indirectly, the measurement relationship may be nonlinear, motivating an extended or unscented Kalman filter. They also point to affine term-structure models: for Vasicek and CIR short rates, bond yields can provide observations that support a linear Kalman-filter approach. The document gives references rather than a derivation or worked estimation procedure, so it does not specify implementation choices, data requirements, or how to assess parameter uncertainty.
Key ideas
- The OU process models mean reversion with constant diffusion, while CIR-style dynamics allow state-dependent volatility.
- The suitable filtering method depends on whether the state is observed directly or through measurements.
- Nonlinear observation relationships can call for extended or unscented Kalman filtering.
- Affine bond-pricing relationships can make linear Kalman filtering applicable to Vasicek and CIR term-structure models.
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# Parameter estimation of Ornstein–Uhlenbeck and CIR processes
# 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?
## Answer by david (score 3)
https://quant.stackexchange.com/a/9395
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.
## Answer by Vince (score -2)
https://quant.stackexchange.com/a/7958
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.