A. Ahdida and A. Alfonsi, Exact and high-order discretization schemes for Wishart processes and their affine extensions, The Annals of Applied Probability, vol.23, issue.3, pp.1025-1073, 2013.
DOI : 10.1214/12-AAP863

URL : https://hal.archives-ouvertes.fr/hal-00491371

J. I. Allen, M. Eknes, and G. Evensen, An Ensemble Kalman Filter with a complex marine ecosystem model: hindcasting phytoplankton in the Cretan Sea, Annales Geophysicae, vol.21, issue.1, pp.399-411, 2003.
DOI : 10.5194/angeo-21-399-2003

URL : https://hal.archives-ouvertes.fr/hal-00316987

J. L. Anderson, An Ensemble Adjustment Kalman Filter for Data Assimilation, Monthly Weather Review, vol.129, issue.12, pp.2884-2903, 2001.
DOI : 10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.5.9952

J. L. Anderson, A Local Least Squares Framework for Ensemble Filtering, Monthly Weather Review, vol.131, issue.4, pp.634-642, 2003.
DOI : 10.1175/1520-0493(2003)131<0634:ALLSFF>2.0.CO;2

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.6543

J. L. Van-hemmen and T. Ando, An inequality for trace ideals, Communications in Mathematical Physics, vol.16, issue.2, pp.143-148, 1980.
DOI : 10.1007/BF01212822

P. J. Antsaklis and A. N. Michel, A Linear Systems Primer, Birkhäuser, 2007.

O. E. Barndorff-nielsen and R. Stelzer, Positive-definite matrix processes of finite variation, Probability and Mathematical Statistics, pp.3-43, 2007.

A. N. Bishop and P. , Del Moral and A. Niclas. An introduction to Wishart matrix moments, 2017.

A. N. Bishop and P. , On the stability of Kalman-Bucy diffusion processes. arXiv e-print, 2017.

A. N. Bishop, P. , and S. Pathiraja, Perturbations and Projections of Kalman-Bucy Semigroups. arXiv e-print, 2017.

M. W. Browne, Asymptotically distribution-free methods for the analysis of covariance structures, British Journal of Mathematical and Statistical Psychology, vol.37, issue.1, pp.62-83, 1984.
DOI : 10.1111/j.2044-8317.1984.tb00789.x

M. F. Bru, Wishart processes, Journal of Theoretical Probability, vol.20, issue.4, pp.725-751, 1991.
DOI : 10.1007/BF01259552

M. F. Bru, Diffusions of perturbed principal component analysis, Journal of Multivariate Analysis, vol.29, issue.1, pp.127-136, 1989.
DOI : 10.1016/0047-259X(89)90080-8

URL : http://doi.org/10.1016/0047-259x(89)90080-8

R. S. Bucy, Global theory of the Riccati equation, Journal of Computer and System Sciences, vol.1, issue.4, pp.349-361, 1967.
DOI : 10.1016/S0022-0000(67)80025-4

G. Burgers, P. J. Van-leeuwen, and G. Evensen, Analysis Scheme in the Ensemble Kalman Filter, Monthly Weather Review, vol.126, issue.6, pp.1719-1724, 1998.
DOI : 10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2

C. Cuchiero, D. Filipovic, E. Mayerhofer, and J. Teichmann, Affine processes on positive semidefinite matrices. The Annals of Applied Probability, pp.397-463, 2011.
DOI : 10.2139/ssrn.1481151

URL : http://arxiv.org/abs/0910.0137

P. and D. Moral, Mean field simulation for Monte Carlo integration, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00932211

P. , D. Moral, and A. Niclas, A Taylor expansion of the square root matrix functional. arXiv e-print, 2017.

P. , D. Moral, and J. Tugaut, On the stability and the uniform propagation of chaos properties of ensemble Kalman-Bucy filters, to appear in the Annals of Applied Probability, 2016.

P. , D. 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, issue.1, pp.119-155, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01469198

P. , D. Moral, and S. Penev, Stochastic Processes: From Applications to Theory, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01593827

G. Evensen, The Ensemble Kalman Filter: theoretical formulation and practical implementation . Ocean Dynamics, pp.343-367, 2003.
DOI : 10.1007/s10236-003-0036-9

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.5.6990

G. Evensen, J. Hove, H. C. Meisingset, E. Reiso, and K. S. Seim, Using the EnKF for Assisted History Matching of a North Sea Reservoir Model, SPE Reservoir Simulation Symposium, 2007.
DOI : 10.2118/106184-MS

C. Gourieroux, J. Jasiak, and R. Sufana, The Wishart Autoregressive process of multivariate stochastic volatility, Journal of Econometrics, vol.150, issue.2, pp.167-181, 2009.
DOI : 10.1016/j.jeconom.2008.12.016

P. Graczyk and L. Vostrikova, Moments of Wishart Processes via Ito Calculus. Theory of Probability & Its Applications, pp.609-625, 2007.
DOI : 10.4213/tvp22

A. Graham, Kronecker Products and Matrix Calculus with Applications, J. Wiley & Sons, 1981.

N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008.
DOI : 10.1137/1.9780898717778

R. Horn and C. R. Johnson, Topics in Matrix Analysis, 1985.
DOI : 10.1017/CBO9780511840371

P. L. Houtekamer and H. L. Mitchell, Data Assimilation Using an Ensemble Kalman Filter Technique, Monthly Weather Review, vol.126, issue.3, pp.796-811, 1998.
DOI : 10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2

URL : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.3.1706&rep=rep1&type=pdf

Y. Hu and X. Y. Zhou, Indefinite Stochastic Riccati Equations, SIAM Journal on Control and Optimization, vol.42, issue.1, pp.123-137, 2003.
DOI : 10.1137/S0363012901391330

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, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, pp.1563-1576, 2011.
DOI : 10.1016/j.jmaa.2005.04.052

C. J. Johns and J. Mandel, A two-stage ensemble Kalman filter for smooth data assimilation, Environmental and Ecological Statistics, vol.131, issue.9, pp.101-110, 2008.
DOI : 10.1080/01621459.1989.10478794

URL : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.129.3631&rep=rep1&type=pdf

E. Kalnay, Atmospheric Modelling, Data Assimilation, and Predictability, 2003.
DOI : 10.1017/cbo9780511802270

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.88.5725

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 1996.

M. Katori and H. Tanemura, Noncolliding Brownian motions and Harish-Chandra formula, Electronic Communications in Probability, vol.8, issue.0, pp.112-121, 2003.
DOI : 10.1214/ECP.v8-1076

URL : http://arxiv.org/abs/math/0306386

M. Katori and H. Tanemura, Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems, Journal of Mathematical Physics, vol.37, issue.8, pp.3058-3085, 2004.
DOI : 10.1063/1.531675

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, C. Griffin & Company, 1943.

E. De-klerk, Aspects of Semidefinite Programming, The Netherlands, vol.65, 2002.
DOI : 10.1007/b105286

K. J. Law, H. Tembine, and R. Tempone, Deterministic Mean-Field Ensemble Kalman Filtering, SIAM Journal on Scientific Computing, vol.38, issue.3, pp.1251-1279, 2016.
DOI : 10.1137/140984415

URL : http://arxiv.org/pdf/1409.0628

F. Le-gland, V. Monbet, and V. D. Tran, Large sample asymptotics for the ensemble Kalman filter. Chapter 22 in The Oxford Handbook of Nonlinear Filtering, pp.598-631, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00744737

K. A. Lisaeter, J. Rosanova, and G. Evensen, Assimilation of ice concentration in a coupled iceocean model using the Ensemble Kalman Filter. Ocean Dynamics, pp.368-388, 2003.

A. J. Majda and J. Harlim, Filtering Complex Turbulent Systems, 2012.
DOI : 10.1017/CBO9781139061308

A. J. Majda and X. T. Tong, Robustness and accuracy of finite ensemble Kalman filters in large dimensions, 2016.

J. Mandel, L. Cobb, and J. D. Beezley, On the convergence of the ensemble Kalman filter, Applications of Mathematics, vol.56, issue.6, pp.533-541, 2011.
DOI : 10.1007/s10492-011-0031-2

E. Mayerhofer, O. Pfaffel, and R. Stelzer, On strong solutions for positive definite jumpdiffusions, Stochastic Processes and Their Applications, pp.2072-2086, 2011.
DOI : 10.1016/j.spa.2011.05.006

URL : http://doi.org/10.1016/j.spa.2011.05.006

H. P. Mckean, A CLASS OF MARKOV PROCESSES ASSOCIATED WITH NONLINEAR PARABOLIC EQUATIONS, Proceedings of the National Academy of Sciences, pp.1907-1911, 1966.
DOI : 10.1073/pnas.56.6.1907

S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models Probabilistic Models for Nonlinear Partial Differential Equations, part of the Lecture Notes in Mathematics book series (LNM, pp.42-95, 1996.

G. Naevdal, L. M. Johnsen, S. I. Aanonsen, and E. H. Vefring, Reservoir Monitoring and Continuous Model Updating Using Ensemble Kalman Filter, Proceedings of the 2003 SPE Annual Technical Conference and Exhibition, 2003.
DOI : 10.2118/84372-PA

E. Ott, B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich et al., A local ensemble Kalman filter for atmospheric data assimilation. Tellus A, pp.415-428, 2004.

A. Seiler, G. Evensen, J. Skjervheim, J. Hove, and J. G. Vab, Using the EnKF for history matching and uncertainty quantification of complex reservoir models, Large-Scale Inverse Problems and Quantification of Uncertainty, pp.247-271, 2010.

A. S. Sznitman, Topics in propagation of chaos Course given at the Ecole d'Eté de Probabilités de Saint-Flour in 1989, part of the Lecture Notes in Mathematics book series (LNM, pp.164-251, 1991.

X. T. Tong, A. J. Majda, and D. Kelly, Nonlinear stability and ergodicity of ensemble based Kalman filters, Nonlinearity, vol.29, issue.2, pp.657-691, 2016.
DOI : 10.1088/0951-7715/29/2/657

X. T. Tong, A. J. Majda, and D. Kelly, Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation, Communications in Mathematical Sciences, vol.14, issue.5, pp.1283-1313, 2016.
DOI : 10.4310/CMS.2016.v14.n5.a5

C. F. Van-loan, The ubiquitous Kronecker product, Journal of Computational and Applied Mathematics, vol.123, issue.1-2, pp.85-100, 2000.
DOI : 10.1016/S0377-0427(00)00393-9