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Affine processes with compact state space. (English) Zbl 1390.60277
Summary: The behavior of affine processes, which are ubiquitous in a wide range of applications, depends crucially on the choice of state space. We study the case where the state space is compact, and prove in particular that (i) no diffusion is possible; (ii) jumps are possible and enforce a grid-like structure of the state space; (iii) jump components can feed into drift components, but not vice versa. Using our main structural theorem, we classify all bivariate affine processes with compact state space. Unlike the classical case, the characteristic function of an affine process with compact state space may vanish, even in very simple cases.

60J25 Continuous-time Markov processes on general state spaces
60J27 Continuous-time Markov processes on discrete state spaces
60J75 Jump processes (MSC2010)
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[1] C. Carathéodory, über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), no. 1, 95–115. · JFM 38.0448.01
[2] C. Cuchiero, Affine and polynomial processes, Ph.D. thesis, ETH ZURICH, 2011.
[3] Christa Cuchiero, Damir Filipović, Eberhard Mayerhofer, and Josef Teichmann, Affine processes on positive semidefinite matrices, Ann. Appl. Probab. 21 (2011), no. 2, 397–463. · Zbl 1219.60068
[4] Christa Cuchiero, Martin Keller-Ressel, Eberhard Mayerhofer, and Josef Teichmann, Affine processes on symmetric cones, J. Theoret. Probab. 29 (2016), no. 2, 359–422. · Zbl 1342.60125
[5] D. Duffie, D. Filipović, and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab. 13 (2003), no. 3, 984–1053. · Zbl 1048.60059
[6] Darrell Duffie, Jun Pan, and Kenneth Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica 68 (2000), no. 6, 1343–1376. · Zbl 1055.91524
[7] Darrell Duffie and Kenneth J Singleton, Modeling term structures of defaultable bonds, Review of Financial studies 12 (1999), no. 4, 687–720.
[8] Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986, Characterization and convergence. · Zbl 0592.60049
[9] Damir Filipović, Term-structure models, Springer Finance, Springer-Verlag, Berlin, 2009, A graduate course.
[10] Damir Filipović and Martin Larsson, Polynomial diffusions and applications in finance, Finance Stoch. 20 (2016), no. 4, 931–972. · Zbl 1386.60237
[11] Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003. · Zbl 1018.60002
[12] Jan Kallsen and Paul Krühner, On a Heath-Jarrow-Morton approach for stock options, Finance Stoch. 19 (2015), no. 3, 583–615. · Zbl 1390.91302
[13] Martin Keller-Ressel and Eberhard Mayerhofer, Exponential moments of affine processes, Ann. Appl. Probab. 25 (2015), no. 2, 714–752. · Zbl 1332.60115
[14] Monika Piazzesi, Affine term structure models, Handbook of financial econometrics 1 (2010), 691–766.
[15] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. · Zbl 0242.90044
[16] Peter Spreij and Enno Veerman, Affine diffusions with non-canonical state space, Stoch. Anal. Appl. 30 (2012), no. 4, 605–641. · Zbl 1260.60112
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