A perturbation analysis of stochastic matrix Riccati diffusions. (English. French summary) Zbl 1434.60020

Summary: Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces.


60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
62M20 Inference from stochastic processes and prediction
60G35 Signal detection and filtering (aspects of stochastic processes)


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[1] A. Ahdida and A. Alfonsi. Exact and high order discretization schemes for Wishart processes and their affine extensions. Ann. Appl. Probab. 23 (3) (2013) 1025-1073. · Zbl 1269.65003
[2] P. J. Antsaklis and A. N. Michel. A Linear Systems Primer. Birkhäuser, Boston, 2007. · Zbl 1168.93001
[3] M. Arnaudon and P. Del Moral A variational approach to nonlinear and interacting diffusions. arXiv preprint, 2018. Available at arXiv:1812.04269.
[4] O. E. Barndorff-Nielsen and R. Stelzer. Positive-definite matrix processes of finite variation. Probab. Math. Statist. 27 (1) (2007) 3-43. · Zbl 1128.60053
[5] K. Bergemann and S. Reich. An ensemble Kalman-Bucy filter for continuous data assimilation. Meteorologische Zeitschrift. 21 (3) (2012) 213-219.
[6] A. N. Bishop and P. Del Moral. On the stability of Kalman-Bucy diffusion processes. SIAM J. Control Optim. 55 (6) (2017) 4015-4047. Available at arXiv:1610.04686. · Zbl 1390.60151
[7] A. N. Bishop and P. Del Moral. On the stability of matrix-valued Riccati diffusions, 2018. Available at arXiv:1808.00235.
[8] A. N. Bishop and P. Del Moral. Stability properties of systems of linear stochastic differential equations with random coefficients. SIAM J. Control Optim. 57 (2) (2019) 1023-1042. Available at arXiv:1804.09349. · Zbl 1415.34101
[9] A. N. Bishop and P. Del Moral. An explicit Floquet-type representation of Riccati aperiodic exponential semigroups. International Journal of Control. Available at arXiv:1805.02127.
[10] A. N. Bishop, P. Del Moral, K. Kamatani and B. Rémillard. On one-dimensional Riccati diffusions. Ann. Appl. Probab. 29 (2) (2019) 1127-1187. · Zbl 1466.60115
[11] A. N. Bishop, P. Del Moral and A. Niclas. An introduction to Wishart matrix moments. Found. Trends Mach. Learn. 11 (2) (2018) 97-218. Available at arXiv:1710.10864. · Zbl 1442.60006
[12] A. N. Bishop, P. Del Moral and S. Pathiraja. Perturbations and projections of Kalman-Bucy semigroups. Stochastic Process. Appl. 128 (9) (2018) 2857-2904. · Zbl 1407.62338
[13] J. M. Bismut. Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 14 (3) (1976) 419-444. · Zbl 0331.93086
[14] M. W. Browne. Asymptotically distribution-free methods for the analysis of covariance structures. Br. J. Math. Stat. Psychol. 37 (1) (1984) 62-83. · Zbl 0561.62054
[15] M. F. Bru. Diffusions of perturbed principal component analysis. J. Multivariate Anal. 29 (1) (1989) 127-136. · Zbl 0687.62048
[16] M. F. Bru. Wishart processes. J. Theoret. Probab. 4 (4) (1991) 725-751. · Zbl 0737.60067
[17] C. Cuchiero, D. Filipovic, E. Mayerhofer and J. Teichmann. Affine processes on positive semi-definite matrices. Ann. Appl. Probab. 21 (2) (2011) 397-463. · Zbl 1219.60068
[18] J. de Wiljes, S. Reich and W. Stannat. Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and small measurement noise. SIAM J. Appl. Dyn. Syst. 17 (2) (2018) 1152-1181. · Zbl 1391.93265
[19] P. Del Moral. Mean Field Simulation for Monte Carlo Integration. Chapman & Hall/CRC Press, London, 2013. · Zbl 1282.65011
[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 J. Control Optim. 55 (1) (2017) 119-155. · Zbl 1356.60065
[21] P. Del Moral and A. Niclas. A Taylor expansion of the square root matrix functional. J. Math. Anal. Appl. 1 (465) (2018) 259-266. Available at arXiv:1705.08561. · Zbl 1401.15009
[22] P. Del Moral and S. Penev. Stochastic Processes: From Applications to Theory. CRC Press-Chapman & Hall, London, 2016. · Zbl 1368.60001
[23] P. Del Moral and J. Tugaut. On the stability and the uniform propagation of chaos properties of ensemble Kalman-Bucy filters. Ann. Appl. Probab. 28 (2) (2018) 790-850. · Zbl 1391.60082
[24] J. L. Doob. Stochastic Processes. J. Wiley & Sons, New York, 1953.
[25] G. Evensen. The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dyn. 53 (4) (2003) 343-367.
[26] C. Gourieroux, J. Jasiak and R. Sufana. The Wishart autoregressive process of multivariate stochastic volatility. J. Econometrics 150 (2) (2009) 167-181. · Zbl 1429.62397
[27] P. Graczyk and L. Vostrikova. Moments of Wishart processes via Ito calculus. Theory Probab. Appl. 51 (4) (2007) 609-625. · Zbl 1131.60008
[28] A. Graham. Kronecker Products and Matrix Calculus with Applications. J. Wiley & Sons, New York, 1981. · Zbl 0497.26005
[29] N. J. Higham. Functions of Matrices: Theory and Computation. SIAM, Philadelphia, 2008. · Zbl 1167.15001
[30] P. L. Houtekamer and H. L. Mitchell. Data assimilation using an ensemble Kalman filter technique. Mon. Weather Rev. 126 (3) (1998) 796-811.
[31] Y. Hu and X. Y. Zhou. Indefinite stochastic Riccati equations. SIAM J. Control Optim. 42 (1) (2003) 123-137. · Zbl 1039.60058
[32] M. Hutzenthaler, A. Jentzen and P. E. Kloeden. Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2130) (2011) 1563-1576. · Zbl 1228.65014
[33] E. Kalnay. Atmospheric Modelling, Data Assimilation, and Predictability. Cambridge University Press, Cambridge, 2003.
[34] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1996. · Zbl 0638.60065
[35] M. Katori and H. Tanemura. Non-colliding Brownian motions and Harish-Chandra formula. Electron. Commun. Probab. 8 (2003) 112-121. · Zbl 1067.82028
[36] M. Katori and H. Tanemura. Symmetry of matrix-valued stochastic processes and non-colliding diffusion particle systems. J. Math. Phys. 45 (8) (2004) 3058-3085. · Zbl 1071.82045
[37] D. T. B. Kelly, K. J. H. Law and A. M. Stuart. Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time. Nonlinearity 27 (10) (2014) 2579-2603. · Zbl 1305.62323
[38] M. G. Kendall and A. Stuart. The Advanced Theory of Statistics. C. Griffin & Company, London, 1943. · Zbl 0416.62001
[39] M. Kohlmann and S. Tang. Multidimensional backward stochastic Riccati equations and applications. SIAM J. Control Optim. 41 (6) (2003) 1696-1721. · Zbl 1175.93242
[40] K. J. H. Law, H. Tembine and R. Tempone. Deterministic mean-field ensemble Kalman filtering. SIAM J. Sci. Comput. 38 (3) (2016) A1251-A1279. · Zbl 1351.60047
[41] F. Le Gland, V. Monbet and V. D. Tran. Large sample asymptotics for the ensemble Kalman filter. In The Oxford Handbook of Nonlinear Filtering 598-631, 2011. · Zbl 1225.93108
[42] A. J. Majda and X. T. Tong. Performance of ensemble Kalman filters in large dimensions, 2016. Available at arXiv:1606.09321. · Zbl 1390.93799
[43] J. Mandel, L. Cobb and J. D. Beezley. On the convergence of the ensemble Kalman filter. Appl. Math. 56 (6) (2011) 533-541. · Zbl 1248.62164
[44] E. Mayerhofer, O. Pfaffel and R. Stelzer. On strong solutions for positive definite jump-diffusions. In Stochastic Processes and Their Applications 2072-2086, 121, 2011. · Zbl 1225.60096
[45] H. P. McKean. A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56 (6) (1966) 1907-1911. · Zbl 0149.13501
[46] S. Méléard. Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. Probabilistic models for nonlinear partial differential equations. In Lecture Notes in Mathematics Book Series (LNM) 42-95, 1627. Springer, Berlin, 1996. · Zbl 0864.60077
[47] S. Reich and C. J. Cotter. Ensemble filter techniques for intermittent data assimilation. In Large Scale Inverse Problems: Computational Methods and Applications in the Earth Sciences 91-134. M. Cullen, M. A. Freitag, S. Kindermann and R. Scheichl (Eds). De Gruyter, Berlin, 2013. Available at arXiv:1208.6572. · Zbl 1291.65032
[48] P. Sakov and P. R. Oke. A deterministic formulation of the ensemble Kalman filter: An alternative to ensemble square root filters. Tellus A. 60 (2) (2008) 361-371.
[49] A. S. Sznitman Topics in Propagation of Chaos. In Course Given at the Ecole d’Eté de Probabilités de Saint-Flour in 1989 Lecture Notes in Mathematics Book Series (LNM) 1464 164-251. Springer-Verlag, Berlin, 1991.
[50] A. Taghvaei and P. G. Mehta. An optimal transport formulation of the linear feedback particle filter. In Proc. of the 2016 American Control Conference (ACC), Boston, USA, 2016.
[51] X. T. Tong, A. J. Majda and D. Kelly. Nonlinear stability and ergodicity of ensemble based Kalman filters. Nonlinearity 29 (2) (2016) 657-691. · Zbl 1334.60060
[52] X. T. Tong, A. J. Majda and D. Kelly. Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation. Commun. Math. Sci. 14 (5) (2016) 1283-1313. · Zbl 1370.37009
[53] J. L. van Hemmen and T. Ando. An inequality for trace ideals. Comm. Math. Phys. 76 (143) (1980) 143-148. · Zbl 0449.47036
[54] C. F. Van Loan. The ubiquitous Kronecker product. J. Comput. Appl. Math. 123 (1) (2000) 85-100. · Zbl 0966.65039
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