26 results on '"Anita Behme"'
Search Results
2. A 2~˟ ~2 random switching model and its dual risk model.
- Author
-
Anita Behme and Philipp Lukas Strietzel
- Published
- 2021
- Full Text
- View/download PDF
3. On moments of downward passage times for spectrally negative Lévy processes
- Author
-
Philipp Lukas Strietzel and Anita Behme
- Subjects
Statistics and Probability ,General Mathematics ,Probability (math.PR) ,FOS: Mathematics ,60G51, 60G40 (primary), 91G05 (secondary) ,Statistics, Probability and Uncertainty - Abstract
The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether the Lévy process is drifting to $+\infty$, $-\infty$ or oscillates. Whenever the Lévy process drifts to $+\infty$, we prove that the $κ$-th moment of the first passage time (conditioned to be finite) exists if and only if the $(κ+1)$-th moment of the Lévy jump measure exists. This generalises a result shown earlier by Delbaen for Cramér-Lundberg risk processes \cite{Delbaen1990}. Whenever the Lévy process drifts to $-\infty$, we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments and in particular we show that no integer moments exist., 15 pages
- Published
- 2022
- Full Text
- View/download PDF
4. On q-scale functions of spectrally negative Lévy processes
- Author
-
Anita Behme, David Oechsler, and René Schilling
- Subjects
Statistics and Probability ,Applied Mathematics - Abstract
We obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.
- Published
- 2022
- Full Text
- View/download PDF
5. Continuity properties and the support of killed exponential functionals
- Author
-
Alexander Lindner, Jana Reker, Victor Rivero, and Anita Behme
- Subjects
Statistics and Probability ,Stationary distribution ,Exponential distribution ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Absolute continuity ,01 natural sciences ,Lévy process ,Exponential function ,010104 statistics & probability ,Stochastic differential equation ,Mathematics::Probability ,Modeling and Simulation ,FOS: Mathematics ,High Energy Physics::Experiment ,0101 mathematics ,Connection (algebraic framework) ,Random variable ,Mathematics - Probability ,Mathematical physics ,Mathematics - Abstract
For two independent Levy processes ξ and η and an exponentially distributed random variable τ with parameter q > 0 , independent of ξ and η , the killed exponential functional is given by V q , ξ , η ≔ ∫ 0 τ e − ξ s − d η s . Interpreting the case q = 0 as τ = ∞ , the random variable V q , ξ , η is a natural generalisation of the exponential functional ∫ 0 ∞ e − ξ s − d η s , the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional V q , ξ , η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of ∫ 0 t e − ξ s − d η s for fixed t ≥ 0 , as well as for integrals of the form ∫ 0 ∞ f ( s ) d η s for deterministic functions f . Furthermore, applying the same techniques to the case q = 0 , new results on the absolute continuity of the improper integral ∫ 0 ∞ e − ξ s − d η s are derived.
- Published
- 2021
- Full Text
- View/download PDF
6. Asymmetric COGARCH processes.
- Author
-
Anita Behme, Claudia Klüppelberg, and Kathrin Mayr
- Published
- 2014
- Full Text
- View/download PDF
7. Moments of MGOU processes and positive semidefinite matrix processes.
- Author
-
Anita Behme
- Published
- 2012
- Full Text
- View/download PDF
8. Moments of the ruin time in a Lévy risk model
- Author
-
Philipp Lukas Strietzel and Anita Behme
- Subjects
Statistics and Probability ,Mathematics::Probability ,60G51, 60G40, 91G05 ,General Mathematics ,Probability (math.PR) ,Mathematics::Optimization and Control ,FOS: Mathematics ,Mathematics - Probability - Abstract
We derive formulas for the moments of the ruin time in a L\'evy risk model and use these to determine the asymptotic behavior of the moments of the ruin time as the initial capital tends to infinity. In the special case of the perturbed Cram\'er-Lundberg model with phase-type or exponentially distributed claims, we explicitly compute the first two moments of the ruin time. All our considerations distinguish between the profitable and the unprofitable setting., Comment: 24 pages, 1 figure
- Published
- 2021
- Full Text
- View/download PDF
9. On the law of killed exponential functionals
- Author
-
Jana Reker, Anita Behme, and Alexander Lindner
- Subjects
Statistics and Probability ,Stationary distribution ,Exponential distribution ,010102 general mathematics ,Probability (math.PR) ,Absolute continuity ,Lebesgue integration ,01 natural sciences ,Lévy process ,Exponential function ,010104 statistics & probability ,symbols.namesake ,Law ,symbols ,FOS: Mathematics ,Infinitesimal generator ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Probability ,Mathematics - Abstract
For two independent L\'{e}vy processes $\xi$ and $\eta$ and an exponentially distributed random variable $\tau$ with parameter $q>0$ that is independent of $\xi$ and $\eta$, the killed exponential functional is given by $V_{q,\xi,\eta} := \int_0^\tau \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$. With the killed exponential functional arising as the stationary distribution of a Markov process, we calculate the infinitesimal generator of the process and use it to derive different distributional equations describing the law of $V_{q,\xi,\eta}$, as well as functional equations for its Lebesgue density in the absolutely continuous case. Various special cases and examples are considered, yielding more explicit information on the law of the killed exponential functional and illustrating the applications of the equations obtained. Interpreting the case $q=0$ as $\tau=\infty$ leads to the classical exponential functional $\int_0^\infty \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$, allowing to extend many previous results to include killing.
- Published
- 2020
10. Markov-modulated generalized Ornstein-Uhlenbeck processes and an application in risk theory
- Author
-
Anita Behme and Apostolos Sideris
- Subjects
Statistics and Probability ,Mathematics::Probability ,Probability (math.PR) ,FOS: Mathematics ,Mathematics - Probability - Abstract
We derive the Markov-modulated generalized Ornstein-Uhlenbeck process by embedding a Markov-modulated random recurrence equation in continuous time. The obtained process turns out to be the unique solution of a certain stochastic differential equation driven by a bivariate Markov-additive process. We present this stochastic differential equation as well as its solution explicitely in terms of the driving Markov-additive process. Moreover, we give necessary and sufficient conditions for strict stationarity of the Markov-modulated generalized Ornstein-Uhlenbeck process, and prove that its stationary distribution is given by the distribution of a specific exponential functional of Markov-additive processes. Finally we propose an application of the Markov-modulated generalized Ornstein-Uhlenbeck process as Markov-modulated risk model with stochastic investment. This generalizes Paulsen's risk process to a Markov-switching environment. We derive a formula in this risk model that expresses the ruin probability in terms of the distribution of an exponential functional of a Markov-additive process.
- Published
- 2020
- Full Text
- View/download PDF
11. Theory of Stochastic Objects: Probability, Stochastic Processes and Inference
- Author
-
Anita Behme
- Subjects
Statistics and Probability ,Probability theory ,Computer science ,business.industry ,Stochastic process ,Inference ,Artificial intelligence ,Statistics, Probability and Uncertainty ,business ,Task (project management) - Abstract
The task of writing a novel textbook on general probability theory seems virtually impossible nowadays. In Theory of Stochastic Objects, the author Athanasios Micheas aims to overcome this challeng...
- Published
- 2020
- Full Text
- View/download PDF
12. Ruin probabilities for risk processes in a bipartite network
- Author
-
Anita Behme, Claudia Klüppelberg, Gesine Reinert, and Lehrstuhl für Mathematische Statistik
- Subjects
Statistics and Probability ,Multivariate statistics ,Mathematics::Optimization and Control ,Poisson distribution ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics::Probability ,Risk process ,FOS: Mathematics ,0101 mathematics ,ddc:510 ,Mathematics ,Bipartite network, Cramér–Lundberg model, exponential claim size distribution, hitting probability, multivariate compound Poisson process, ruin theory, Poisson approximation, Pollaczek–Khintchine formula, risk balancing network ,Actuarial science ,Applied Mathematics ,Risk balancing ,010102 general mathematics ,Probability (math.PR) ,Insurance market ,Ruin theory ,Modeling and Simulation ,Mathematik ,Bipartite graph ,symbols ,Mathematics - Probability - Abstract
This paper studies risk balancing features in an insurance market by evaluating ruin probabilities for single and multiple components of a multivariate compound Poisson risk process. The dependence of the components of the process is induced by a random bipartite network. In analogy with the non-network scenario, a network ruin parameter is introduced. This random parameter, which depends on the bipartite network, is crucial for the ruin probabilities. Under certain conditions on the network and for light-tailed claim size distributions we obtain Lundberg bounds and, for exponential claim size distributions, exact results for the ruin probabilities. For large sparse networks, the network ruin parameter is approximated by a function of independent Poisson variables. T
- Published
- 2019
13. Exponential functionals of Markov additive processes
- Author
-
Anita Behme and Apostolos Sideris
- Subjects
Statistics and Probability ,Markov process ,01 natural sciences ,Lévy process ,Exponential integral ,010104 statistics & probability ,symbols.namesake ,60H10, 60J75, 60G51, 60J25 ,60J25 ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics ,Markov chain ,Markov additive process ,010102 general mathematics ,Probability (math.PR) ,exponential functional ,Exponential function ,Markov modulated perpetuity ,Markov switching model ,Convergence of random variables ,symbols ,60H10 ,Statistics, Probability and Uncertainty ,60J75 ,Mathematics - Probability ,60G51 - Abstract
We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. By contrast with the classical Levy case studied by Erickson and Maller we have to distinguish between almost sure convergence and convergence in probability. Our proofs rely on recent results on perpetuities in a Markovian environment by Alsmeyer and Buckmann.
- Published
- 2019
- Full Text
- View/download PDF
14. A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions
- Author
-
Anita Behme and Lennart Bondesson
- Subjects
Statistics and Probability ,Class (set theory) ,Lévy process ,010102 general mathematics ,excursion theory ,products and ratios of independent random variables ,Scale (descriptive set theory) ,0102 computer and information sciences ,hyperbolic monotonicity ,01 natural sciences ,Combinatorics ,Monotone polygon ,Integer ,010201 computation theory & mathematics ,Order (group theory) ,exponential functionals ,0101 mathematics ,Random variable ,Mathematics ,generalized gamma convolution - Abstract
Let $k>0$ be an integer and $Y$ a standard $\operatorname{Gamma}(k)$ 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 ($\mathrm{GGC}$s). This result extends a result of Roynette et al. from 2009 who treated the case $k=1$ but without use of the $\mathrm{HM}$-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Lévy processes are mentioned.
- Published
- 2017
15. Séminaire de Probabilités XLVIII
- Author
-
Mathias Beiglböck, Martin Huesmann, Makoto Maejima, Nicolas Privault, Franck Maunoury, Anna Aksamit, Ismael Bailleul, Nicolas Juillet, David Applebaum, Christophe Profeta, Dai Taguchi, Peter Kern, Kilian Raschel, Alexis Devulder, Libo Li, Camille Tardif, Anita Behme, Thomas Simon, Gilles Pagès, Florian Stebegg, Oleskiy Khorunzhiy, Jürgen Angst, Stéphane Laurent, Cédric Lecouvey, Songzi Li, Wendelin Werner, Alexander Lindner, Matyas Barczy, Vienna University of Technology (TU Wien), Rheinische Friedrich-Wilhelms-Universität Bonn, Columbia University [New York], Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Institute of Mathematics [Debrecen], University of Debrecen Egyetem [Debrecen], Technische Universität Dresden = Dresden University of Technology (TU Dresden), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Nanyang Technological University [Singapour], Universität Ulm - Ulm University [Ulm, Allemagne], Keio University, Laboratoire de Mathématiques et Physique Théorique (LMPT), Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et Modélisation d'Evry (LaMME), Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), North Dakota State University (NDSU), Laboratoire Traitement et Communication de l'Information (LTCI), Télécom ParisTech-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Mathematical Institute [Oxford] (MI), University of Oxford, Université d'Évry-Val-d'Essonne (UEVE), School of Mathematics and Statistics [Sheffield] (SoMaS), University of Sheffield [Sheffield], Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Donati-Martin, Catherine, Lejay, Antoine, Rouault, Alain, ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Technische Universität Wien (TU Wien), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Université de Versailles Saint-Quentin-en-Yvelines (UVSQ), University of Debrecen, Technische Universität Dresden (TUD), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Université de Tours-Centre National de la Recherche Scientifique (CNRS), Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé - UMR 8524 (LPP), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), University of Oxford [Oxford], Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology in Zürich [Zürich] (ETH Zürich), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire de Mathématiques et Modélisation d'Evry, Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Technische Universität Wien ( TU Wien ), Bonn Universität [Bonn], Institut de Recherche Mathématique Avancée ( IRMA ), Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Probabilités et Modèles Aléatoires ( LPMA ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), TO Simulate and CAlibrate stochastic models ( TOSCA ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut Élie Cartan de Lorraine ( IECL ), Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques de Versailles ( LMV ), Université Paris-Saclay-Centre National de la Recherche Scientifique ( CNRS ) -Université de Versailles Saint-Quentin-en-Yvelines ( UVSQ ), Technische Universität Dresden ( TUD ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Universität Ulm, Laboratoire de Mathématiques et Physique Théorique ( LMPT ), Université de Tours-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques et Modélisation d'Evry ( LaMME ), Institut National de la Recherche Agronomique ( INRA ) -Université d'Évry-Val-d'Essonne ( UEVE ) -ENSIIE-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Paul Painlevé - UMR 8524 ( LPP ), Université de Lille-Centre National de la Recherche Scientifique ( CNRS ), North Dakota State University ( NDSU ), Laboratoire Traitement et Communication de l'Information ( LTCI ), Télécom ParisTech, Institut Fourier ( IF ), Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Mathematical Institute [Oxford] ( MI ), Université d'Évry-Val-d'Essonne ( UEVE ), School of Mathematics and Statistics [Sheffield] ( SoMaS ), and Eidgenössische Technische Hochschule [Zürich] ( ETH Zürich )
- Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,[ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR] ,ComputingMilieux_MISCELLANEOUS - Abstract
International audience
- Published
- 2016
- Full Text
- View/download PDF
16. On the Range of Exponential Functionals of Lévy Processes
- Author
-
Alexander Lindner, Anita Behme, and Makoto Maejima
- Subjects
010104 statistics & probability ,Range (mathematics) ,Weak convergence ,010102 general mathematics ,Mathematical analysis ,Compound Poisson process ,0101 mathematics ,01 natural sciences ,Lévy process ,Brownian motion ,Exponential function ,Mathematics ,Mathematical physics - Abstract
We characterize the support of the law of the exponential functional \(\int _{0}^{\infty }e^{-\xi _{s-}}\,d\eta _{s}\) of two one-dimensional independent Levy processes ξ and η. Further, we study the range of the mapping Φ ξ for a fixed Levy process ξ, which maps the law of η1 to the law of the corresponding exponential functional \(\int _{0}^{\infty }e^{-\xi _{s-}}\,d\eta _{s}\). 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.
- Published
- 2016
- Full Text
- View/download PDF
17. Multivariate generalized Ornstein–Uhlenbeck processes
- Author
-
Alexander Lindner and Anita Behme
- Subjects
Statistics and Probability ,Discrete mathematics ,Multiplicative Lévy process ,Lévy process ,Applied Mathematics ,Invariant subspace ,Irreducible model ,Ornstein–Uhlenbeck process ,Generalized Ornstein–Uhlenbeck process ,Linear subspace ,Combinatorics ,Stochastic differential equation ,Modelling and Simulation ,Modeling and Simulation ,Stochastic exponential ,Irreducibility ,Invariant (mathematics) ,Random variable ,Mathematics - Abstract
De Haan and Karandikar (1989) [7] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes ( V t ) t ≥ 0 which are basically characterized by the fact that for each h > 0 the equidistantly sampled process ( V n h ) n ∈ N 0 satisfies the random recurrence equation V n h = A ( n − 1 ) h , n h V ( n − 1 ) h + B ( n − 1 ) h , n h , n ∈ N , where ( A ( n − 1 ) h , n h , B ( n − 1 ) h , n h ) n ∈ N is an i.i.d. sequence with positive A 0 , h for each h > 0 . We generalize this concept to a multivariate setting and use it to define multivariate generalized Ornstein–Uhlenbeck (MGOU) processes which occur to be characterized by a starting random variable and some Levy process ( X , Y ) in R m × m × R m . The stochastic differential equation an MGOU process satisfies is also derived. We further study invariant subspaces and irreducibility of the models generated by MGOU processes and use this to give necessary and sufficient conditions for the existence of strictly stationary MGOU processes under some extra conditions.
- Published
- 2012
- Full Text
- View/download PDF
18. Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise
- Author
-
Anita Behme
- Subjects
Statistics and Probability ,Applied Mathematics ,010102 general mathematics ,Autocorrelation ,Mathematical analysis ,Process (computing) ,Bivariate analysis ,01 natural sciences ,Lévy process ,010104 statistics & probability ,Stochastic differential equation ,Levy noise ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.
- Published
- 2011
- Full Text
- View/download PDF
19. Stationary solutions of the stochastic differential equation dVt=Vt−dUt+dLt with Lévy noise
- Author
-
Ross Maller, Alexander Lindner, and Anita Behme
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,Mathematical analysis ,Lévy process ,Exponential function ,Stochastic differential equation ,Levy noise ,Measurement theory ,Distribution (mathematics) ,Modeling and Simulation ,Filtration (mathematics) ,Stationary solution ,Mathematics - Abstract
For a given bivariate Levy process ( U t , L t ) t ≥ 0 , necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation d V t = V t − d U t + d L t are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V 0 and ( U , L ) is assumed and non-causal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for U and L to remain semimartingales with respect to the corresponding expanded filtration is given.
- Published
- 2011
- Full Text
- View/download PDF
20. Laplace Symbols and Invariant Distributions
- Author
-
Alexander Schnurr and Anita Behme
- Subjects
Statistics and Probability ,Pure mathematics ,60G10 (primary), 60G51, 60J35, 47G30 (secondary) ,Laplace transform ,010102 general mathematics ,Probability (math.PR) ,01 natural sciences ,010104 statistics & probability ,Bounded function ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Infinitesimal generator ,0101 mathematics ,Statistics, Probability and Uncertainty ,Invariant (mathematics) ,Mathematics - Probability ,Mathematics - Abstract
We introduce a new kind of symbol in the framework of It\^o 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., Comment: 10 pages
- Published
- 2015
- Full Text
- View/download PDF
21. A criterion for invariant measures of Itô processes based on the symbol
- Author
-
Anita Behme, Alexander Schnurr, and Lehrstuhl für Mathematische Statistik
- Subjects
Statistics and Probability ,invariant measure ,Feller process, Invariant measure, Itô process, Lévy-type process, Stationarity, Stochastic differential equation, Symbol ,Lévy-type process ,Probabilistic logic ,Markov process ,stochastic differential equation ,Feller process ,Itô process ,symbol ,symbols.namesake ,Stochastic differential equation ,stationarity ,Mathematik ,symbols ,Applied mathematics ,Invariant measure ,Invariant (mathematics) ,ddc:510 ,Ito process ,Mathematics - Probability ,Mathematics - Abstract
An integral criterion for the existence of an invariant measure of an It\^{o} process is developed. This new criterion is based on the probabilistic symbol of the It\^{o} 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., Comment: Published at http://dx.doi.org/10.3150/14-BEJ618 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
- Published
- 2014
22. On exponential functionals of Lévy processes
- Author
-
Alexander Lindner, Anita Behme, and Lehrstuhl für Mathematische Statistik
- Subjects
Statistics and Probability ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Probability (math.PR) ,Inverse ,Feller process ,Lévy process ,Injective function ,Exponential function ,Range (mathematics) ,Mathematics::Probability ,Mathematik ,FOS: Mathematics ,Infinitesimal generator ,Statistics, Probability and Uncertainty ,ddc:510 ,60G10, 60G51, 60J35 ,Mathematics - Probability ,Mathematics - Abstract
Exponential functionals of Levy 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 $$\Phi $$ , which maps two independent Levy 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 (Levy) characteristics. Also, continuity of $$\Phi $$ is treated, and some results on its range are obtained.
- Published
- 2014
23. Asymmetric COGARCH processes
- Author
-
Kathrin Mayr, Claudia Klüppelberg, and Anita Behme
- Subjects
maximum-likelihood estimation ,Statistics and Probability ,60G10, 60G51, 62M05 ,Autoregressive conditional heteroskedasticity ,Maximum likelihood ,General Mathematics ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Method of moments (statistics) ,01 natural sciences ,010104 statistics & probability ,62M05 ,0502 economics and business ,FOS: Mathematics ,Applied mathematics ,stochastic volatility ,0101 mathematics ,Mathematics ,Estimation ,COGARCH ,method of moments ,050208 finance ,GJR-GARCH ,Stochastic volatility ,05 social sciences ,Probability (math.PR) ,90G70 ,asymmetric power COGARCH ,APCOGARCH ,high-frequency data ,Moment (mathematics) ,Econometric model ,first-jump approximation ,Jump ,continuous-time GARCH ,62M10 ,Statistics, Probability and Uncertainty ,GJR-COGARCH ,60G10 ,60G51 ,62F10 ,Mathematics - Probability - Abstract
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., Comment: 15 pages, 1 figure
- Published
- 2014
- Full Text
- View/download PDF
24. Superposition of COGARCH processes
- Author
-
Anita Behme, Claudia Klüppelberg, Carsten Chong, and Lehrstuhl für Mathematische Statistik
- Subjects
Statistics and Probability ,primary: 60G10 secondary: 60G51, 60G57, 60H05 ,Stochastic volatility ,Applied Mathematics ,Probability (math.PR) ,Lévy process ,Random measure ,Autocovariance ,Superposition principle ,Modeling and Simulation ,Mathematik ,Jump ,Econometrics ,COGARCH, continuous-time GARCH model, independently scattered, infinite divisibility, Lévy basis, Lévy process, random measure, stationarity, stochastic volatility process, supCOGARCH, superposition ,FOS: Mathematics ,Financial volatility ,Volatility (finance) ,ddc:510 ,Mathematics - Probability ,Mathematics - Abstract
We suggest three superpositions of COGARCH (supCOGARCH) volatility processes driven by L\'evy processes or L\'evy 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., Comment: 41 pages, 6 figures
- Published
- 2013
- Full Text
- View/download PDF
25. Distributions of exponential integrals of independent increment processes related to generalized gamma convolutions
- Author
-
Makoto Maejima, Anita Behme, Noriyoshi Sakuma, and Muneya Matsui
- Subjects
Statistics and Probability ,Pure mathematics ,Lévy process ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,generalized gamma convolutions ,Convolution ,Exponential integral ,exponential integral ,Mathematics::Probability ,selfdecomposable distribution ,FOS: Mathematics ,Mathematics - Abstract
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., Published in at http://dx.doi.org/10.3150/11-BEJ382 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
- Published
- 2012
26. Exponential Functionals of Lévy Processes with Jumps
- Author
-
Anita Behme and Lehrstuhl für Mathematische Statistik
- Subjects
Mathematik ,COGARCH volatility, exponential functional, generalized gamma convolution, generalized Ornstein-Uhlenbeck process, integral mapping, Levy process, selfdecomposability, stationarity ,ddc:510
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.