arXiv subjects
Publications and source records attributed to Anita Behme.
We introduce a new kind of symbol in the framework of Itô processes which are bounded on one side. The connection between this symbol and the infinitesimal generator is analyzed. Based on this concept, an integral criterion for invariant distributions of the underlying process is derived. Some applications are mentioned.
Let $ k >0 $ be an integer and $ Y $ a standard Gamma$(k)$ distributed random variable. Let $ X $ be an independent positive random variable with a density that is hyperbolically monotone (HM) of order $ k.$ Then $Y\cdot X$ and $Y/X $ both have distributions that are generalized gamma convolutions (GGCs). This result extends a result of Roynette et al. from 2009 who treated the case $ k=1 $ but without use of the HM-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Lévy processes are mentioned.
An integral criterion for the existence of an invariant measure of an Itô process is developed. This new criterion is based on the probabilistic symbol of the Itô process. In contrast to the standard integral criterion for invariant measures of Markov processes based on the generator, no test functions and hence no information on the domain of the generator is needed.
We study the exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$ of two one-dimensional independent Lévy processes $ξ$ and $η$, where $η$ is a subordinator. In particular, we derive an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping $Φ_ξ$ for a fixed Lévy process $ξ$, which maps the law of $η_1$ to the law of the corresponding exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$, and study the behaviour of the range of $Φ_ξ$ for varying characteristics of $ξ$. Moreover, we derive conditions for selfdecomposable distributions and generalized Gamma convolutions to be in the range. On the way we also obtain new characterizations of these classes of distributions.
We suggest three superpositions of COGARCH (supCOGARCH) volatility processes driven by Lévy processes or Lévy bases. We investigate second-order properties, jump behaviour, and prove that they exhibit Pareto-like tails. Corresponding price processes are defined and studied. We find that the supCOGARCH models allow for more flexible autocovariance structures than the COGARCH. Moreover, other than most financial volatility models, the supCOGARCH processes do not exhibit a deterministic relationship between price and volatility jumps. Furthermore, in one supCOGARCH model not all volatility jumps entail a price jump, while in another supCOGARCH model not all price jumps necessarily lead to volatility jumps.
We characterize the support of the law of the exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$ of two one-dimensional independent Lévy processes $ξ$ and $η$. Further, we study the range of the mapping $Φ_ξ$ for a fixed Lévy process $ξ$, which maps the law of $η_1$ to the law of the corresponding exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$. It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.
Financial data are as a rule asymmetric, although most econometric models are symmetric. This applies also to continuous-time models for high-frequency and irregularly spaced data. We discuss some asymmetric versions of the continuous-time GARCH model, concentrating then on the GJR-COGARCH. We calculate higher order moments and extend the first jump approximation. These results are prerequisites for moment estimation and pseudo maximum likelihood estimation of the GJR-COGARCH parameters, respectively, which we derive in detail.
Exponential functionals of Lévy processes appear as stationary distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further we use these results to investigate properties of the mapping Φ, which maps two independent Lévy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective and give the inverse mapping in terms of (Lévy) characteristics. Also, continuity of Φis treated and some results on its range are obtained.
It is known that in many cases distributions of exponential integrals of Levy processes are infinitely divisible and in some cases they are also selfdecomposable. In this paper, we give some sufficient conditions under which distributions of exponential integrals are not only selfdecomposable but furthermore are generalized gamma convolution. We also study exponential integrals of more general independent increment processes. Several examples are given for illustration.