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Abstract (English)

This thesis is devoted to the study of ergodic properties of some one and two-dimensional affine processes. Roughly speaking, the class of affine processes on the canonical state space, introduced by Duffie, Filipović, and Schachermayer (2013), consists of continuous-time Markov processes taking values in R_(≥0)^m×R^n whose log-characteristic function depends in an affine way on the initial state vector of the process. A question of interest in the context of time-homogeneous Markov processes is their long-time behavior such as the ergodicity. Until now, ergodicity is not very well investigated for general affine processes. This is one reason why we initially started to work on particular (non-trivial) affine models such as a jump-type Cox-Ingersoll-Ross process and a two-factor model based on the α-root process. A further reason is given by the fact that both models discussed in this thesis provide interesting applications in financial mathematics. In the first part of this thesis we study an affine two-factor model based on the α-root process introduced by Barczy, Döring, Li, and Pap (2014). One component of this two-dimensional model is the so-called α-root process. We manage to prove exponential ergodicity of this two-factor model when α∈(1,2) mainly by stochastic methods, e.g. a Foster-Lyapunov drift criteria developed by Meyn and Tweedie (1993). As a further result of our considerations, we obtain existence of positive transition densities of the α-root process. In the second part of the thesis we introduce the jump-diffusion Cox-Ingersoll-Ross process, which is an extension of the Cox-Ingersoll-Ross model and whose jumps are introduced by a subordinator. We provide sufficient conditions on the Lévy measure of the subordinator under which the jump-diffusion Cox-Ingersoll-Ross process is ergodic and exponentially ergodic, respectively. Furthermore, we characterize the existence of the κ-moment (κ>0) of the jump-diffusion Cox-Ingersoll-Ross process by an integrability condition on the Lévy measure of the subordinator. As a consequence of our results, we obtain a moment convergence theorem for the jump-diffusion Cox-Ingersoll-Ross process. Eventually, to illustrate the use of our ergodic results, we study asymptotic properties of conditional least squares estimators for the drift parameters of the jump-diffusion Cox-Ingersoll-Ross process based on discrete time observations. In the subcritical case we prove strong consistency and asymptotic normality of our parameter estimators.

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