The Annals of Probability

Skew convolution semigroups and affine Markov processes

D. A. Dawson and Zenghu Li

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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.

Article information

Ann. Probab., Volume 34, Number 3 (2006), 1103-1142.

First available in Project Euclid: 27 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H20: Stochastic integral equations 60K37: Processes in random environments

Skew convolution semigroup affine process continuous state branching process catalytic branching process immigration Ornstein–Uhlenbeck process stochastic integral equation Poisson random measure


Dawson, D. A.; Li, Zenghu. Skew convolution semigroups and affine Markov processes. Ann. Probab. 34 (2006), no. 3, 1103--1142. doi:10.1214/009117905000000747.

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