Electronic Journal of Probability

On the stability of matrix-valued Riccati diffusions

Adrian N. Bishop and Pierre Del Moral

Full-text: Open access


The stability properties of matrix-valued Riccati diffusions are investigated. The matrix-valued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. Under rather natural observability and controllability conditions, we derive time-uniform moment and fluctuation estimates and exponential contraction inequalities. Our approach combines spectral theory with nonlinear semigroup methods and stochastic matrix calculus. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation. This class of stochastic models arise in signal processing and data assimilation, and more particularly in ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion represents the flow of the sample covariance matrices associated with McKean-Vlasov-type interacting Kalman-Bucy filters. The analysis developed here applies to filtering problems with unstable signals.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 84, 40 pp.

Received: 29 January 2019
Accepted: 13 July 2019
First available in Project Euclid: 10 September 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G52: Stable processes 93E11: Filtering [See also 60G35] 60G99: None of the above, but in this section

ensemble filtering methods ensemble Kalman-Bucy filters matrix quadratic stochastic differential equations matrix Riccati diffusion equations matrix Riccati equations uniform fluctuation estimates uniform stability estimates

Creative Commons Attribution 4.0 International License.


Bishop, Adrian N.; Del Moral, Pierre. On the stability of matrix-valued Riccati diffusions. Electron. J. Probab. 24 (2019), paper no. 84, 40 pp. doi:10.1214/19-EJP342. https://projecteuclid.org/euclid.ejp/1568080863

Export citation


  • [1] J.L. Anderson and S.L. Anderson. A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Monthly Weather Review. vol. 127, no. 12. pp. 2741–2758 (1999).
  • [2] G.W. Anderson, A. Guionnet, and O. Zeitouni. An Introduction to Random Matrices. Cambridge University Press (2010).
  • [3] B.D.O. Anderson and J.B. Moore. Linear Optimal Control. Prentice-Hall (1971).
  • [4] B.D.O. Anderson and J.B. Moore. Optimal Filtering. Dover Publications (1979).
  • [5] M. Arnaudon and A. Thalmaier. The differentiation of hypoelliptic diffusion semigroups. Illinois Journal of Mathematics. vol. 54, no. 4. pp. 1285–1311 (2010).
  • [6] D.R. Bell. Stochastic differential equations and hypoelliptic operators. In: Rao M.M. (eds) Real and Stochastic Analysis. Trends in Mathematics. pp. 9–42. Birkhüser Boston (2004).
  • [7] D.S. Bernstein. Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press (2005).
  • [8] A.N. Bishop and P. Del Moral. On the stability of Kalman-Bucy diffusion processes. SIAM Journal on Control and Optimization. vol. 55, no. 6. pp 4015–4047 (2017). updated at arXiv e-print, arXiv:1610.04686.
  • [9] A.N. Bishop and P. Del Moral. Stability properties of systems of linear stochastic differential equations with random coefficients. SIAM Journal on Control and Optimization. vol. 57, no. 2. pp. 1023–1042 (2019). arXiv e-print, arXiv:1804.09349 (2018).
  • [10] A.N. Bishop and P. Del Moral. An explicit Floquet-type representation of Riccati aperiodic exponential semigroups. International Journal of Control. doi:10.1080/00207179.2019.1590647. arXiv e-print, arXiv:1805.02127 (2018).
  • [11] A.N. Bishop, P. Del Moral, K. Kamatani, R. Rémillard. On one-dimensional Riccati diffusions. Annals of Applied Probability. vol. 29, no. 2. pp. 1127–1187 (2019). arXiv e-print, arXiv:1711.10065 (2017).
  • [12] A.N. Bishop, P. Del Moral, and A. Niclas. A perturbation analysis of stochastic matrix Riccati diffusions. Annales de l’Institut Henri Poincaré: Probab. & Statist. to appear; arXiv e-print, arXiv:1709.05071 (2017).
  • [13] A.N. Bishop, P. Del Moral and S. Pathiraja. Perturbations and projections of Kalman-Bucy semigroups. Stochastic Processes and their Applications. vol. 128, no. 9. pp. 2857–2904. (2018).
  • [14] J.M. Bismut. Linear quadratic optimal stochastic control with random coefficients. SIAM Journal on Control and Optimization. vol. 14, no. 3. pp. 419–444 (1976).
  • [15] M. Bramanti. An Invitation to Hypoelliptic Operators and Hörmander’s Vector Fields. Springer (2014).
  • [16] M.F. Bru. Wishart processes. Journal of Theoretical Probability. vol. 4, no. 4. pp. 725–751 (1991).
  • [17] F.M. Callier and J.L. Willems. Criterion for the convergence of the solution of the Riccati differential equation. IEEE Transactions on Automatic Control. vol. 26, no. 6. pp. 1232–1242 (1981).
  • [18] J.C. Cox, J.E. Ingersoll and S.A. Ross. A theory of the term structure of interest rate. Econometrica. vol. 53, no. 2. pp. 385–407 (1985).
  • [19] C. Cuchiero, D. Filipovich, E. Mayerhofer, and J. Teichman. Affine processes on positive semidefinite matrices. The Annals of Applied Probability. vol. 21, no. 2. pp. 397–463 (2011)
  • [20] P. Del Moral, A. Kurtzmann, and J. Tugaut. On the stability and the uniform propagation of chaos of a class of extended ensemble Kalman–Bucy filters. SIAM Journal on Control and Optimization. vol. 55, no. 1. pp. 119–155 (2017).
  • [21] P. Del Moral and S. Penev. Stochastic Processes: From Applications to Theory. CRC Press (2017).
  • [22] P. Del Moral and S.S. Singh. A forward-backward stochastic analysis of diffusion flows. arXiv e-print, arXiv:1906.09145 (2019).
  • [23] P. Del Moral and J. Tugaut. On the stability and the uniform propagation of chaos properties of ensemble Kalman-Bucy filters. Annals of Applied Probability. vol. 28, no. 2. pp 790–850 (2018).
  • [24] J.L. Doob. Stochastic Processes. J. Wiley & Sons, New York (1953).
  • [25] F.J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. Journal of Mathematical Physics. vol. 3, no. 6. pp. 1191–1198 (1962).
  • [26] G. Evensen. The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dynamics. vol. 53, no. 4. pp. 343–367 (2003).
  • [27] M. Fiedler. Special Matrices and Their Applications in Numerical Mathematics. 2nd Edition. Dover Publications (2008).
  • [28] S. Friedland. Variation of tensor powers and spectra. Linear and Multilinear Algebra. vol. 12, no. 2. pp. 81–98 (1982).
  • [29] M. Hairer. Convergence of Markov processes. LectureNotes. University of Warwick (January 2016).
  • [30] T.M. Hamill, J.S. Whitaker, and C. Snyder. Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Monthly Weather Review. vol. 129, no. 11. pp. 2776–2790 (2001).
  • [31] U.G. Haussmann and E. Pardoux. A conditionally almost linear filtering problem with non-Gaussian initial condition. Stochastics. vol. 23, no. 2. pp. 241–275 (1988).
  • [32] Y. Hu and X.Y. Zhou. Indefinite stochastic Riccati equations. SIAM Journal on Control Optimization. vol. 42, no. 1. pp. 123–137 (2003).
  • [33] I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus. Springer (1996).
  • [34] M. Keller-Ressel, W. Schachermayer and J. Teichmann. Affine processes are regular. Probability Theory and Related Fields. vol. 151, no. 3–4. pp. 591–611 (2011).
  • [35] R. Khasminskii. Stochastic Stability of Differential Equations. Springer Science & Business Media (2011).
  • [36] M. Kohlmann and S. Tang. Multidimensional backward stochastic Riccati equations and applications. SIAM Journal on Control Optimization. vol. 41, no. 6. pp. 1696–1721 (2003).
  • [37] G.M. Krause. Bounds for the variation of matrix eigenvalues and polynomial roots. Linear Algebra and its Applications. vol. 208-209. pp. 73–82 (1994).
  • [38] V. Kucera. A contribution to matrix quadratic equations. IEEE Transactions on Automatic Control. vol. 17, no. 3. pp. 344–347 (1972).
  • [39] H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley-Interscience (1972).
  • [40] P. Lancaster and L. Rodman. Algebraic Riccati Equations. Oxford University Press (1995).
  • [41] R.S. Liptser and A.N. Shiryaev. Statistics of Random Processes (Vol. 1 and 2). 2nd Edition. Springer-Verlag (2001).
  • [42] E. Mayerhofer, O. Pfaffel, and R. Stelzer. On strong solutions for positive definite jump-diffusions. Stochastic Processes and Their Applications. vol. 121, no. 9. pp. 2072–2086 (2011).
  • [43] M.L. Mehta. Random Matrices. 3rd Edition. Elsevier/Academic Press (2004).
  • [44] B.P. Molinari. The time-invariant linear-quadratic optimal control problem. Automatica. vol. 13, no. 4. pp. 347–357 (1977).
  • [45] P. Sakov and P.R. Oke. A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters. Tellus A. vol. 60, no. 2. pp. 361-371 (2008).
  • [46] B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M.I. Jordan and S.S. Sastry. Kalman filtering with intermittent observations. IEEE Transactions on Automatic Control. vol. 49, no. 9. pp. 1453–1464 (2004).
  • [47] T. Tao. Topics in Random Matrix Theory. American Mathematical Society (2012).