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Affine processes on symmetric cones. (English) Zbl 1342.60125
Summary: We consider affine Markov processes taking values in convex cones. In particular, we characterize all affine processes taking values in irreducible symmetric cones in terms of certain Lévy-Khintchine triplets. This is the natural, coordinate-free formulation of the theory of Wishart processes on positive semidefinite matrices, as put forward by M.-F. Bru [ibid. 4, No. 4, 725–751 (1991; Zbl 0737.60067)] and C. Cuchiero et al. [Ann. Appl. Probab. 21, No. 2, 397–463 (2011; Zbl 1219.60068)], in the more general context of symmetric cones, which also allows for simpler, alternative proofs.

MSC:
60J25 Continuous-time Markov processes on general state spaces
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[1] Aliprantis, C.D., Tourky, R.: Cones and Duality, volume 84 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2007) · Zbl 1127.46002
[2] Barndorff-Nielsen, OE; Stelzer, R, Positive-definite matrix processes of finite variation, Probab. Math. Statist., 27, 3-43, (2007) · Zbl 1128.60053
[3] Bru, M-F, Wishart processes, J. Theor. Probab., 4, 725-751, (1991) · Zbl 0737.60067
[4] Cuchiero, C., Teichmann, J.: Path Properties and Regularity of Affine Processes on General State Spaces. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds.) Lecture Notes in Mathematics. Séminaire de Probabilités XLV, vol. 2078, pp. 201-244. Springer, Cham (2013) · Zbl 1287.60091
[5] Cuchiero, C; Filipović, D; Mayerhofer, E; Teichmann, J, Affine processes on positive semidefinite matrices, Ann. Appl. Probab., 21, 397-463, (2011) · Zbl 1219.60068
[6] Damm, T, Positive groups on \({\cal {H}}^n\) are completely positive, Linear Algebra Appl., 393, 127-137, (2004) · Zbl 1073.47044
[7] Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1969). Enlarged and corrected printing, Pure and Applied Mathematics, Vol. 10-I · Zbl 0176.00502
[8] Duffie, D; Filipović, D; Schachermayer, W, Affine processes and applications in finance, Ann. Appl. Probab., 13, 984-1053, (2003) · Zbl 1048.60059
[9] Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford Science Publications, New York (1994) · Zbl 0841.43002
[10] Grasselli, M; Tebaldi, C, Solvable affine term structure models, Math. Financ., 18, 135-153, (2008) · Zbl 1138.91547
[11] Gowda, M; Sznajder, R; Tao, J, The automorphism group of a completely positive cone and its Lie algebra, Linear Algebra Appl., 438, 3862-3871, (2013) · Zbl 1286.22007
[12] Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I, volume 8 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (1993). Nonstiff problems · Zbl 0789.65048
[13] Hertneck, C, Positivitätsbereiche und Jordan-strukturen, Math. Ann., 146, 433-455, (1962) · Zbl 0143.05202
[14] Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex analysis and minimization algorithms. I, volume 305 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1993). Fundamentals
[15] Ishi, H, The gradient maps associated to certain non-homogeneous cones, Proc. Jpn. Acad. Ser. A Math. Sci., 81, 44-46, (2005) · Zbl 1086.52501
[16] Keller-Ressel, M.: Affine processes—theory and applications in mathematical finance. PhD thesis, Vienna University of Technology (2009) · Zbl 1260.60112
[17] Keller-Ressel, M; Schachermayer, W; Teichmann, J, Affine processes are regular, Probab. Theory Relat. Fields, 151, 591-611, (2011) · Zbl 1235.60093
[18] Keller-Ressel, M., Schachermayer, W., Teichmann, J.: Regularity of affine processes on general state spaces. Electron. J. Probab. 18(43), 1-17 (2013) · Zbl 1291.60153
[19] Kuzma, B., Omladič, M., Šivic, K., Teichmann, J.: Exotic one-parameter semigroups of endomorphisms of a symmetric cone. cone (2014). arXiv:1408.2967 · Zbl 1237.60032
[20] Letac, G., Massam, H.: A tutorial on non-central Wishart distributions (2004) · Zbl 1221.60074
[21] Lévy, P, The arithmetic character of the Wishart distribution, Proc. Camb. Philos. Soc., 44, 295-297, (1948) · Zbl 0030.40602
[22] Massam, H; Neher, E, On transformations and determinants of Wishart variables on symmetric cones, J. Theor. Probab., 10, 867-902, (1997) · Zbl 0890.60016
[23] Mayerhofer, E, Affine processes on positive semidefinite \(d× d\) matrices have jumps of finite variation in dimension \(d{\>}1\), Stoch. Process. Their Appl., 122, 3445-3459, (2012) · Zbl 1256.60023
[24] Mayerhofer, E; Muhle-Karbe, J; Smirnov, AG, A characterization of the martingale property of exponentially affine processes, Stoch. Process. Their Appl., 121, 568-582, (2011) · Zbl 1237.60032
[25] Mayerhofer, E; Pfaffel, O; Stelzer, R, On strong solutions for positive definite jump-diffusions, Stoch. Process. Their Appl., 121, 2072-2086, (2011) · Zbl 1225.60096
[26] Muirhead, R.: Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1982) · Zbl 0556.62028
[27] Nomura, T, Manifold of primitive idempotents in a Jordan-Hilbert algebra, J. Math. Soc. Jpn., 45, 37-58, (1993) · Zbl 0791.58011
[28] Pigorsch, C; Stelzer, R, On the definition, stationary distribution and second order structure of positive semidefinite Ornstein-Uhlenbeck type processes, Bernoulli, 15, 754-773, (2009) · Zbl 1221.60074
[29] Sato, K.: Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990). Translated from the 1990 Japanese original, Revised by the author
[30] Skorohod, A.V.: Random Processes with Independent Increments, volume 47 of Mathematics and Its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1991). Translated from the second Russian edition by P. V. Malyshev
[31] Spreij, P; Veerman, E, Affine diffusions with non-canonical state space, Stoch. Anal. Their Appl., 30, 605-641, (2012) · Zbl 1260.60112
[32] Veerman, E.: Affine Markov processes on a general Euclidean state space. PhD Thesis (2011)
[33] Vinberg, ÈB, Homogeneous cones, Sov. Math. Dokl., 1, 787-790, (1960) · Zbl 0143.05203
[34] Volkmann, P, Über die invarianz konvexer mengen und differentialungleichungen in einem normierten raume, Math. Ann., 203, 201-210, (1973) · Zbl 0251.34039
[35] Walter, W.: Gewöhnliche Differentialgleichungen. Springer-Lehrbuch. [Springer Textbook], 5th edn. Springer, Berlin (1993). Eine Einführung. [An introduction] · Zbl 0772.34001
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