Large deviations for wishart processes

• Autorzy: Donati-Martin C.
• Języki publikacji: EN

Abstrakty

EN

Let X^δ be a Wishart process of dimension δ, with values in the set of positive matrices of size m. We are interested in the large deviations for a family of matrix-valued processes {δ⁻¹X^(δ)_t ; t ≤1} as δ tends to infinity. The process X^(δ) is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.

Słowa kluczowe

EN

Wishart process; large deviation principle

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2008
• Tom: Vol. 28, Fasc. 2
• Strony: 325--343
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Donati-Martin C.

• UPMC Université Paris 6, CNRS UMR 7599, Laboratoire de Probabilités et Modèles Aléatoires, Site Chevaleret, 16 rue Clisson, F-75013 Paris, France

Bibliografia

• [1] M.-F. Bru, Wishart processes, J. Theoret. Probab. 4 (1991), pp. 725-751.
• [2] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd edition, Springer 1998.
• [3] J.-D. Deuschel and D.W. Stroock, Large Deviations, Pure and Applied Mathematics 137, Academic Press, Inc., Boston, MA, 1989.
• [4] C. Donati-Martin, Y. Doumerc, H. Matsumoto and M. Yor, Some properties of the Wishart processes and a matrix extension of the Hartman-Watson laws, Publ. Res. Inst. Math. Sci. 40 (2004), pp. 1385-1412.
• [5] C. Donati-Martin, A. Rouault, M. Yor, M. and M. Zani, Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes, Probab. Theory Related Fields 129 (2004), pp. 261-289.
• [6] S. Feng, The behaviour near the boundary of some degenerate diffusions under random perturbations, in: Stochastic Models (Ottawa, ON, 1998), CMS Conf. Proc. 26, Amer. Math. Soc., Providence, RI, 2000, pp. 115-123.
• [7] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, New York 1984.
• [8] D. G. Luenberger, Optimization by Vector Space Methods, Wiley, 1969.
• [9] H. P. McKean, The Bessel motion and a singular integral equation, Mem. Coll. Sci. Univ. Kyoto. Ser. A, Math. 33 (1960), pp. 317-322.
• [10] J. Pitman and M. Yor, A decomposition of Bessel bridges, Z. Wahrsch. Verw. Gebiete 59 (1982), pp. 425-457.
• [11] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd edition, Springer, Berlin 1999.
• [12] D. W. Stroock and S. R. S Varadhan, Multidimensional Diffusion Processes, Springer, Berlin-New York 1979.