Skew convolution semigroups and affine Markov processes
D. A. Dawson and Zenghu Li
Source: Ann. Probab. Volume 34, Number 3
(2006), 1103-1142.
Abstract
A general affine Markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup. We provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup. The corresponding affine Markov process is constructed as the strong solution of a system of stochastic equations with non-Lipschitz coefficients and Poisson-type integrals over some random sets. Based on this characterization, it is proved that the affine process arises naturally in a limit theorem for the difference of a pair of reactant processes in a catalytic branching system with immigration.
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Keywords: Skew convolution semigroup; affine process; continuous state branching process; catalytic branching process; immigration; Ornstein–Uhlenbeck process; stochastic integral equation; Poisson random measure
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1151418494
Digital Object Identifier: doi:10.1214/009117905000000747
Mathematical Reviews number (MathSciNet): MR2243880
Zentralblatt MATH identifier: 1102.60065
References
Aldous, D. (1978). Stopping times and tightness. Ann. Probab. 6 335--340.
Mathematical Reviews (MathSciNet): MR0474446
Digital Object Identifier: doi:10.1214/aop/1176995579
Project Euclid: euclid.aop/1176995579
Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1406564
Bogachev, V. I., Röckner, M. and Schmuland, B. (1996). Generalized Mehler semigroups and applications. Probab. Theory Related Fields 105 193--225.
Mathematical Reviews (MathSciNet): MR1392452
Dawson, D. A., Etheridge, A. M., Fleischmann, K., Mytnik, L., Perkins, E. A. and Xiong, J. (2002). Mutually catalytic branching in the plane: Finite measure states. Ann. Probab. 30 1681--1762.
Mathematical Reviews (MathSciNet): MR1944004
Digital Object Identifier: doi:10.1214/aop/1039548370
Project Euclid: euclid.aop/1039548370
Zentralblatt MATH: 1017.60098
Dawson, D. A. and Fleischmann, K. (1997). A continuous super-Brownian motion in a super-Brownian medium. J. Theoret. Probab. 10 213--276.
Mathematical Reviews (MathSciNet): MR1432624
Digital Object Identifier: doi:10.1023/A:1022606801625
Zentralblatt MATH: 0877.60030
Dawson, D. A. and Fleischmann, K. (2000). Catalytic and mutually catalytic branching. In Infinite Dimensional Stochastic Analysis (Amsterdam, 1999) 145--170. Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. 52. R. Neth. Acad. Arts Sci., Amsterdam.
Mathematical Reviews (MathSciNet): MR1831416
Zentralblatt MATH: 0983.92036
Dawson, D. A. and Li, Z. H. (2004). Non-differentiable skew convolution semigroups and related Ornstein--Uhlenbeck processes. Potential Anal. 20 285--302.
Mathematical Reviews (MathSciNet): MR2032499
Digital Object Identifier: doi:10.1023/B:POTA.0000010666.75722.0e
Zentralblatt MATH: 1037.60071
Dawson, D. A., Li, Z. H., Schmuland, B. and Sun, W. (2004). Generalized Mehler semigroups and catalytic branching processes with immigration. Potential Anal. 21 75--97.
Mathematical Reviews (MathSciNet): MR2048508
Digital Object Identifier: doi:10.1023/B:POTA.0000021337.13730.8c
Zentralblatt MATH: 1057.60074
Dellacherie, C. and Meyer, P. A. (1982). Probabilites and Potential. Chapters V--VIII. North-Holland, Amsterdam.
Mathematical Reviews (MathSciNet): MR0745449
Zentralblatt MATH: 0494.60002
Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Probab. 13 984--1053.
Mathematical Reviews (MathSciNet): MR1994043
Digital Object Identifier: doi:10.1214/aoap/1060202833
Project Euclid: euclid.aoap/1060202833
Zentralblatt MATH: 1048.60059
Dynkin, E. B., Kuznetsov, S. E. and Skorokhod, A. V. (1994). Branching measure-valued processes. Probab. Theory Related Fields 99 55--96.
Mathematical Reviews (MathSciNet): MR1273742
Digital Object Identifier: doi:10.1007/BF01199590
Zentralblatt MATH: 0813.60076
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
Mathematical Reviews (MathSciNet): MR0838085
Zentralblatt MATH: 0592.60049
Fang, S. and Zhang, T. (2005). A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Related Fields 132 356--390.
Mathematical Reviews (MathSciNet): MR2197106
Digital Object Identifier: doi:10.1007/s00440-004-0398-z
Zentralblatt MATH: 1081.60043
Fuhrman, M. and Röckner, M. (2000). Generalized Mehler semigroups: The non-Gaussian case. Potential Anal. 12 1--47.
Mathematical Reviews (MathSciNet): MR1745332
Digital Object Identifier: doi:10.1023/A:1008644017078
Zentralblatt MATH: 0957.47028
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland/Kodansha, Amsterdam/Tokyo.
Mathematical Reviews (MathSciNet): MR1011252
Zentralblatt MATH: 0684.60040
Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 34--51.
Mathematical Reviews (MathSciNet): MR0290475
Laha, R. G. and Rohatgi, V. K. (1979). Probability Theory. Wiley, New York.
Mathematical Reviews (MathSciNet): MR0534143
Zentralblatt MATH: 0409.60001
Li, Z. H. (1996). Convolution semigroups associated with measure-valued branching processes. Chinese Sci. Bull. (English Edition) 41 276--280.
Mathematical Reviews (MathSciNet): MR1398983
Li, Z. H. (1996). Immigration structures associated with Dawson--Watanabe superprocesses. Stochastic Process. Appl. 62 73--86.
Mathematical Reviews (MathSciNet): MR1388763
Digital Object Identifier: doi:10.1016/0304-4149(95)00087-9
Zentralblatt MATH: 0847.60071
Li, Z. H. (1998). Immigration processes associated with branching particle systems. Adv. in Appl. Probab. 30 657--675.
Mathematical Reviews (MathSciNet): MR1663513
Digital Object Identifier: doi:10.1239/aap/1035228122
Project Euclid: euclid.aap/1035228122
Zentralblatt MATH: 0926.60066
Li, Z. H. (2002). Skew convolution semigroups and related immigration processes. Theory Probab. Appl. 46 274--296.
Mathematical Reviews (MathSciNet): MR1968685
van Neerven, J. M. A. M. (2000). Continuity and representation of Gaussian--Mehler semigroups. Potential Anal. 13 199--211.
Mathematical Reviews (MathSciNet): MR1800373
Digital Object Identifier: doi:10.1023/A:1008799213878
Zentralblatt MATH: 0974.47036
Rhyzhov, Y. M. and Skorokhod, A. V. (1970). Homogeneous branching processes with a finite number of types and continuous varying mass. Theory Probab. Appl. 15 704--707.
Mathematical Reviews (MathSciNet): MR0288860
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1739520
Sharpe, M. J. (1988). General Theory of Markov Processes. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR0958914
Zentralblatt MATH: 0649.60079
Schmuland, B. and Sun, W. (2001). On the equation $\mu_t+s = \mu_s * T_s\mu_t$. Statist. Probab. Lett. 52 183--188.
Mathematical Reviews (MathSciNet): MR1841407
Shiga, T. and Watanabe, S. (1973). Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete 27 37--46.
Mathematical Reviews (MathSciNet): MR0368192
Digital Object Identifier: doi:10.1007/BF00736006
Watanabe, S. (1969). On two dimensional Markov processes with branching property. Trans. Amer. Math. Soc. 136 447--466.
Mathematical Reviews (MathSciNet): MR0234531
Digital Object Identifier: doi:10.2307/1994726
Zentralblatt MATH: 0209.19303
Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 155--167.
Mathematical Reviews (MathSciNet): MR0278420
