Affine processes are regular

Abstract

Abstract. We show that stochastically continuous, time-homogeneous affine processes on the canonical state space R m 0 × R n are always regular. In the paper of Duffie, Filipovic, and Schachermayer (2003) regularity was used as a crucial basic assumption. It was left open whether this regularity condition is automatically satisfied, for stochastically continuous affine processes. We now show that the regularity assumption is indeed superfluous, since regularity follows from stochastic continuity and the exponentially affine behavior of the characteristic function. For the proof we combine classic results on the differentiability of transformation semigroups with the method of the moving frame which has been recently found to be useful in the theory of SPDEs.

Stationarity and Ergodicity for an Affine Two-Factor Model

“…Note also that the process (Y t ) t 0 given by the first SDE of (1.1) is a continuous state branching process with immigration with branching mechanism bz + 1 α z α , z 0, and with immigration mechanism az, z 0 (for more details, see the proof of Theorem 3.1, part (i)). Chen and Joslin [7] have found several applications of the model (1.1) with α = 2 in financial mathematics, see their equations (25) and (26).…”

“…A precise mathematical formulation and a complete characterization of regular affine processes are due to Duffie et al [12]. Later several authors have contributed to the theory of general affine processes: to name a few, Andersen and Piterbarg [1], Dawson and Li [11], Filipović and Mayerhofer [13], Glasserman and Kim [16], Jena et al [22] and Keller-Ressel et al [26].…”

Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion

“…It was shown in [2] (see also [13]) that the semigroup of any stochastically continuous affine process on the canonical state space R m + × R n is a Feller semigroup. Define the semigroup of the BAJD by…”

“…Roughly speaking, affine processes are Markov processes for which the logarithm of the characteristic function of the process is affine with respect to the initial state. Affine processes on the canonical state space R m + × R n have been thoroughly investigated by Duffie et al [2], as well as in [13]. In particular, it was shown in [2] (see also [13]) that any stochastic continuous affine process on R m + × R n is a Feller process and a complete characterization of its generator has been derived.…”

“…Affine processes on the canonical state space R m + × R n have been thoroughly investigated by Duffie et al [2], as well as in [13]. In particular, it was shown in [2] (see also [13]) that any stochastic continuous affine process on R m + × R n is a Feller process and a complete characterization of its generator has been derived. Results on affine processes with the state space R + can also be found in [4].…”