zbMATH — the first resource for mathematics

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.

60J25 Continuous-time Markov processes on general state spaces
Full Text: DOI arXiv
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.