Skew convolution semigroups and affine Markov processes

By Donald A. Dawson, Zenghu Li. 2005

180 citations recorded by the source corpus.

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.

Subfields: Management Information Systems, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e7ba95fa66f2b9eb31d6483f79d4afc67fcd5ac3

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Citations in the corpus

Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model (2013, methodology): “The proofs depend heavily on the construction and characterization of CBIprocesses in terms of stochastic equations of given in Dawson and Li (2006, 2012), Fu and Li (2010) and Li and Ma (2008).”
Strong solutions of jump-type stochastic equations (2012, unknown): “, [5, 6, 7, 14].”
Proof(s) of the Lamperti representation of Continuous-State Branching Processes (2008, methodology): “The study of the Lamperti transformation uses martingales (as a substitute for the delicate use of generators), and the inverse is analyzed in the spirit of [9] where stochastic diﬀerential equations are shown to be satisﬁed by aﬃne processes, which make explicit in the special case of CSBPs.”
Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration (2014, result): “2) to dimension 1 and 2, respectively, which enables us to compare our results with those of Dawson and Li (2006, Theorems 5.1 and 5.2) (single-type) and Ma (2013, Theorem 2.1) (two-type), respectively. Moreover, we discuss a special case of the SDE (3.2) with ν = 0, μi = 0, i ∈ {1, . . . , d}, i.e.”
Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct (2007, background): “The definition of the constant c comes from the necessary agreement between the last display and (7).”
On strong solutions for positive definite jump diffusions (2009, unknown): “On the other hand, one knows that in the case d = 1 pathwise uniqueness holds, hence there exists strong solutions for all b 0 (even in the general setting of CBI processes, see [ 15, Theorem 5.1 ]).”
Continuous-State Branching Processes with Immigration (2019, background): “Under the first moment assumption, the regularity of affine processes was proved in Dawson and Li (2006).”
MOMENT EXPLOSIONS AND LONG‐TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS (2008, methodology): “The key idea of our proof is originally due to Montgomery and Zippin (1955), and has also been presented in Filipović and Teichmann (2003) and Dawson and Li (2006).”
Stochastic equations of non-negative processes with jumps (2008, methodology): “By an easy application of Theorem 6.1 we get the following result, which improves slightly a theorem of Dawson and Li (2006).”