Electronic Journal of Statistics

Inference for a mean-reverting stochastic process with multiple change points

Fuqi Chen, Rogemar Mamon, and Matt Davison

Full-text: Open access


The use of an Ornstein-Uhlenbeck (OU) process is ubiquitous in business, economics and finance to capture various price processes and evolution of economic indicators exhibiting mean-reverting properties. The time at which structural transition representing drastic changes in the economic dynamics occur are of particular interest to policy makers, investors and financial product providers. This paper addresses the change-point problem under a generalised OU model and investigates the associated statistical inference. We propose two estimation methods to locate multiple change points and show the asymptotic properties of the estimators. An informational approach is employed in detecting the change points, and the consistency of our methods is also theoretically demonstrated. Estimation is considered under the setting where both the number and location of change points are unknown. Three computing algorithms are further developed for implementation. The practical applicability of our methods is illustrated using simulated and observed financial market data.

Article information

Electron. J. Statist., Volume 11, Number 1 (2017), 2199-2257.

Received: April 2016
First available in Project Euclid: 23 May 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G20: Generalized stochastic processes 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 62M99: None of the above, but in this section

Ornstein-Uhlenbeck process sequential analysis least sum of squared errors maximum likelihood consistent estimator segment neighbourhood search method PELT algorithm

Creative Commons Attribution 4.0 International License.


Chen, Fuqi; Mamon, Rogemar; Davison, Matt. Inference for a mean-reverting stochastic process with multiple change points. Electron. J. Statist. 11 (2017), no. 1, 2199--2257. doi:10.1214/17-EJS1282. https://projecteuclid.org/euclid.ejs/1495504915

Export citation


  • Aalen, O., Gjessing, H., Survival models based on the Ornstein-Uhlenbeck process, Lifetime Data Analysis. 10(4), (2004), 407–423.
  • Akaike, H., Information theory and an extension of the maximum likelihood principle, in Petrov, B., Csáki, F., 2nd International Symposium on Information Theory, Tsahkadsor, Armenia, USSR, September 2–8, 1971, Budapest: Akadémiai Kiadó, (1973), 267–281.
  • Auger, I., Lawrence, C., Algorithms for the optimal identification of segment neighborhoods, Bulletin of Mathematical Biology. 51(1), (1989), 39–54.
  • Bai, J., Perron, P., Estimating and testing linear models with multiple structural changes, Econometrica. 66(1), (1998), 47–78.
  • Benth, F., Koekebakker, S., Taib, C., Stochastic dynamical modelling of spot freight rates, IMA Journal of Management Mathematics. 26(3), (2015), 273–297.
  • Chen, S., Modelling the dynamics of commodity prices for investment decisions under uncertainty, PhD Dissertation. University of Waterloo, (2010)., https://uwspace.uwaterloo.ca/bitstream/handle/10012/5504/Chen_Shan.pdf
  • Chen, F., Nkurunziza, S., Optimal method in multiple regression with structural changes, Bernoulli. 21(4), (2015), 2217–2241.
  • Date, P., Bustreo, R., Value-at-risk for fixed-income portfolios: a Kalman filtering approach, IMA Journal of Management Mathematics. 27(4), (2016), 557–573.
  • Date, P., Mamon, R., Tenyakov, A., Filtering and forecasting commodity futures prices under an HMM framework, Energy Economics. 40, (2013), 1001–1013.
  • Dehling, H., Franke, B., Kott, T., Drift estimation for a periodic mean reversion process, Statistical Inference for Stochastic Processes. 13, (2010), 175–192.
  • Dehling, H., Franke, B., Kott, T., Kulperger, R., Change point testing for the drift parameters of a periodic mean reversion process, Statistical Inference for Stochastic Process. 17(1), (2014), 1–18.
  • Ditlevsen, S., Lansky, P., Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model, Physical Review E. 71, (2005), 011907.
  • Elias, R., Wahab, M., Fung, F., A comparison of regime-switching temperature modeling approaches for applications in weather derivatives, European Journal of Operational Research. 232(3), (2014), 549–560.
  • Elliott, R., van der Hoek, J., Malcolm, P., Pairs trading, Quantitative Finance, 5(3) (2005), 271–276.
  • Elliott, P., Wilson, C., The term structure of interest rates in a hidden Markov setting, in Mamon, R., Elliott, R. (eds), Hidden Markov Models in Finance. Springer, New York, (2007), 14–31.
  • Erlwein, C., Benth, F., Mamon, R.S., HMM filtering and parameter estimation of an electricity spot price model, Energy Economics, 32(5), (2010), 1034–1043.
  • Gallagher, C., Lund, R., Robbins, M., Changepoint detection in daily precipitation data, Environmetrics, 23(5), (2012), 407–419.
  • Gombay, E., Change detection in linear regression with time series errors, Canadian Journal of Statistics, 38(1), (2010), 65–79.
  • Hull, J., White, A., Pricing interest-rate-derivative securities, Review of Financial Studies. 3(4), (1990), 573–592.
  • Howell, S., Duck, P., Hazel, A., Johnson, P., Pinto, H., Strbac, G., Proudlove, N., Black, M., A partial differential equation system for modelling stochastic storage in physical systems with applications to wind power generation, IMA Journal of Management Mathematics, 22(3), (2011), 231–252.
  • Jackson, B., Scargle, J., Barnes, D., Arabhi, S., Alt, A., Gioumousis, P., Gwin, E., Sangtrakulcharoen, P., Tan, L., Tsai, T., An algorithm for optimal partitioning of data on an interval, IEEE Signal Processing Letters, 12, (2002), 105–108.
  • Killick, R., Fearnhead, P., Eckley, I., Optimal detection of change points with a linear computational cost, Journal of the American Statistical Association. 107(500), (2012), 1590–1598.
  • Kutoyants, Y.A., Statistical Inference for Ergodic Diffusion Processes, Springer Series in Statistics, London, (2004).
  • Lansky, P., Sacerdote, L., The Ornstein-Uhlenbeck neuronal model with signal-dependent noise, Physics Letters A. 285(3–4), (2001), 132–140.
  • Liang, Z., Yuen, K., Guo, J., Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Mathematics and Economics. 49(2), (2011), 207–215.
  • Lipster, R., Shiryaev, A., Statistics of Random Processes I. Springer-Verlag, Berlin-Heidelber-New York, (2001).
  • Lu, Q., Lund, R., Simple linear regression with multiple level shifts, Canadian Journal of Statistics. 35 (3), (2007), 447–458.
  • Lu, S., Ornstein-Uhlenbeck diffusion quantum Monte Carlo calculations for small first-row polyatomic molecules, Journal of Chemical Physics. 118(21), (2003), 9528–9532.
  • Lu, S., Ornstein-Uhlenbeck diffusion quantum Monte Carlo study on the bond lengths and harmonic frequencies of some first-row diatomic molecules, Journal of Chemical Physics. 120, (2004), http://dx.doi.org/ 10.1063/1.1639370.
  • Maidstone, R., Hocking, T., Rigaill, G., Fearnhead, P., On optimal multiple changepoint algorithms for large data, ArXiv e-prints., (2014).
  • Nkurunziza, S., Zhang, P., Estimation and testing in generalized mean-reverting processes with change-point. Statistical Inference for Stochastic Processes. (2016), in press. DOI, 10.1007/s11203-016-9151-3.
  • Page, E., Continuous inspection schemes, Biometrika. 41, (1954), 100–115.
  • Perron, P., Qu, Z., Estimating restricted structural change models, Journal of Econometrics. 134(2), (2006), 373–399.
  • Reeves, J., Chen, J., Wang, X., Lund, R., Lu, Q., A review and comparison of changepoint detection techniques for climate data, Journal of Applied Meteorology and Climatology. 46(6), (2007), 900–915.
  • Robbins, M., Lund, R., Gallagher, C., Lu, Q., Changepoints in the North Atlantic tropical cyclone record, Journal of the American Statistical Association, 106 (493), (2011), 89–99.
  • Rohlfs, R., Harrigan, P., Nielsen, R., Modeling gene expression evolution with an extended Ornstein-Uhlenbeck process accounting for within-species variation, Scandinavian Journal of Statistics. 37(2), (2010), 200–220.
  • Schwartz, E., The stochastic behavior of commodity prices: implications for valuation and hedging, Journal of Finance 52. (1997), 923–973.
  • Sen, A., Srivastava, M., On tests for detecting change in mean, Annals of Statistics. 3(1), (1975), 98–108.
  • Scott, A., Knott, M., A cluster analysis method for grouping means in the analysis of variance, Biometrics. 30(3), (1974), 507–512.
  • Schwarz, G., Estimating the dimension of a model, Annals of Statistics. 6(2), (1978), 461–464.
  • Shinomoto, S., Sakai, Y., Funahashi, S., The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Computation. 11(4), (1999), 935–951.
  • Shiryaev, A., On optimum methods in quickest detection problems, Theory of Probability and Its Applications. 8, (1963), 26–51.
  • Smith, W., On the simulation and estimation of the mean-reverting Ornstein-Uhlenbeck process. Version 1.01, CommodityModels.com, (2010).
  • Spokoiny, V., Multiscale local change point detection with applications to value-at-risk, Annals of Statistics. 37(3), (2009), 1405–1436.
  • Tenyakov, A., Mamon, R., Davison, M., Modelling high-frequency FX rate dynamics: A zero-delay multi-dimensional HMM-based approach, Knowledge-Based Systems. 101, (2016), 142–155.
  • Tenyakov, A., Mamon, R., Davison, M., Filtering of a discrete-time HMM-driven multivariate Ornstein-Uhlenbeck model with application to forecasting market liquidity regimes, IEEE Journal of Selected Topics in Signal Processing. 10(6), (2016), 994–1005.
  • Tobing, H., McGilchrist, C., Recursive residuals for multivariate regression models, Australian Journal of Statistics. 34(2), (1992), 217–232.
  • Vasiček, O., An equilibrium characterisation of the term structure, Journal of Financial Economics. 5, (1977), 177–188.
  • Yan, G., Xiao, Z., Eidenbenz, S., Catching instant messaging worms with change-point detection techniques, Proceedings of the 1st Usenix Workshop on Large-Scale Exploits and Emergent Threats. 6(8), (2008), 1–10.
  • Yu, X., Baron, M., Choudhary, P., Change-point detection in binomial thinning processes with applications in epidemiology, Sequential Analysis 32 (3), (2013), 350–367.
  • Zhang, P., On Stein-rules in generalized mean-reverting processes with change point, Master’s Thesis, University of Windsor, (2015).