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Probability and Heat Kernel Estimates for Lévy(-Type) ProcessesKühn, Franziska 05 December 2016 (has links) (PDF)
In this thesis, we present a new existence result for Lévy-type processes. Lévy-type processes behave locally like a Lévy process, but the Lévy triplet may depend on the current position of the process. They can be characterized by their so-called symbol; this is the analogue of the characteristic exponent in the Lévy case. Using a parametrix construction, we prove the existence of Lévy-type processes with a given symbol under weak regularity assumptions on the regularity of the symbol. Applications range from existence results for stable-like processes and mixed processes to uniqueness results for Lévy-driven stochastic differential equations.
Moreover, we discuss sufficient conditions for the existence of moments of Lévy-type processes and derive estimates for fractional moments.
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Drift estimation for jump diffusionsMai, Hilmar 08 October 2012 (has links)
Das Ziel dieser Arbeit ist die Entwicklung eines effizienten parametrischen Schätzverfahrens für den Drift einer durch einen Lévy-Prozess getriebenen Sprungdiffusion. Zunächst werden zeit-stetige Beobachtungen angenommen und auf dieser Basis eine Likelihoodtheorie entwickelt. Dieser Schritt umfasst die Frage nach lokaler Äquivalenz der zu verschiedenen Parametern auf dem Pfadraum induzierten Maße. Wir diskutieren in dieser Arbeit Schätzer für Prozesse vom Ornstein-Uhlenbeck-Typ, Cox-Ingersoll-Ross Prozesse und Lösungen linearer stochastischer Differentialgleichungen mit Gedächtnis im Detail und zeigen starke Konsistenz, asymptotische Normalität und Effizienz im Sinne von Hájek und Le Cam für den Likelihood-Schätzer. In Sprungdiffusionsmodellen ist die Likelihood-Funktion eine Funktion des stetigen Martingalanteils des beobachteten Prozesses, der im Allgemeinen nicht direkt beobachtet werden kann. Wenn nun nur Beobachtungen an endlich vielen Zeitpunkten gegeben sind, so lässt sich der stetige Anteil der Sprungdiffusion nur approximativ bestimmen. Diese Approximation des stetigen Anteils ist ein zentrales Thema dieser Arbeit und es wird uns auf das Filtern von Sprüngen führen. Der zweite Teil dieser Arbeit untersucht die Schätzung der Drifts, wenn nur diskrete Beobachtungen gegeben sind. Dabei benutzen wir die Likelihood-Schätzer aus dem ersten Teil und approximieren den stetigen Martingalanteil durch einen sogenannten Sprungfilter. Wir untersuchen zuerst den Fall endlicher Aktivität und zeigen, dass die Driftschätzer im Hochfrequenzlimes die effiziente asymptotische Verteilung erreichen. Darauf aufbauend beweisen wir dann im Falle unendlicher Sprungaktivität asymptotische Effizienz für den Driftschätzer im Ornstein-Uhlenbeck Modell. Im letzten Teil werden die theoretischen Ergebnisse für die Schätzer auf endlichen Stichproben aus simulierten Daten geprüft und es zeigt sich, dass das Sprungfiltern zu einem deutlichen Effizienzgewinn führen. / The problem of parametric drift estimation for a a Lévy-driven jump diffusion process is considered in two different settings: time-continuous and high-frequency observations. The goal is to develop explicit maximum likelihood estimators for both observation schemes that are efficient in the Hájek-Le Cam sense. The likelihood function based on time-continuous observations can be derived explicitly for jump diffusion models and leads to explicit maximum likelihood estimators for several popular model classes. We consider Ornstein-Uhlenbeck type, square-root and linear stochastic delay differential equations driven by Lévy processes in detail and prove strong consistency, asymptotic normality and efficiency of the likelihood estimators in these models. The appearance of the continuous martingale part of the observed process under the dominating measure in the likelihood function leads to a jump filtering problem in this context, since the continuous part is usually not directly observable and can only be approximated and the high-frequency limit. In the second part of this thesis the problem of drift estimation for discretely observed processes is considered. The estimators are constructed from discretizations of the time-continuous maximum likelihood estimators from the first part, where the continuous martingale part is approximated via a thresholding technique. We are able to proof that even in the case of infinite activity jumps of the driving Lévy process the estimator is asymptotically normal and efficient under weak assumptions on the jump behavior. Finally, the finite sample behavior of the estimators is investigated on simulated data. We find that the maximum likelihood approach clearly outperforms the least squares estimator when jumps are present and that the efficiency gap between both techniques becomes even more severe with growing jump intensity.
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Probability and Heat Kernel Estimates for Lévy(-Type) ProcessesKühn, Franziska 25 November 2016 (has links)
In this thesis, we present a new existence result for Lévy-type processes. Lévy-type processes behave locally like a Lévy process, but the Lévy triplet may depend on the current position of the process. They can be characterized by their so-called symbol; this is the analogue of the characteristic exponent in the Lévy case. Using a parametrix construction, we prove the existence of Lévy-type processes with a given symbol under weak regularity assumptions on the regularity of the symbol. Applications range from existence results for stable-like processes and mixed processes to uniqueness results for Lévy-driven stochastic differential equations.
Moreover, we discuss sufficient conditions for the existence of moments of Lévy-type processes and derive estimates for fractional moments.
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Moments of the Ruin Time in a Lévy Risk ModelStrietzel, Philipp Lukas, Behme, Anita 08 April 2024 (has links)
We derive formulas for the moments of the ruin time in a Lévy 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 Cramér-Lundberg model with phase-type or even 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.
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