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1	Lévy-Type Processes under Uncertainty and Related Nonlocal Equations Hollender, Julian 17 October 2016 (has links) (PDF) The theoretical study of nonlinear expectations is the focus of attention for applications in a variety of different fields — often with the objective to model systems under incomplete information. Especially in mathematical finance, advances in the theory of sublinear expectations (also referred to as coherent risk measures) lay the theoretical foundation for modern approaches to evaluations under the presence of Knightian uncertainty. In this book, we introduce and study a large class of jump-type processes for sublinear expectations, which can be interpreted as Lévy-type processes under uncertainty in their characteristics. Moreover, we establish an existence and uniqueness theory for related nonlinear, nonlocal Hamilton-Jacobi-Bellman equations with non-dominated jump terms. Wahrscheinlichkeitstheorie Analysis Stochastische Analysis Stochastische Prozesse Risikomaße Finanzmathematik Risikomaße Sublineare Erwartungswerte Nichtlineare Erwartungswerte Lévy-typ Prozesse Nichtlokale Gleichungen HJB Gleichungen Knightian Uncertainty Probability Theory Analysis Stochastic Analysis Stochastic Processes Risk Measures Mathematical Finance Risk Measures Sublinear Expectation Nonlinear Expectation Lévy-Type Processes Nonlocal Equations HJB Equations Knightian Uncertainty ddc:510 rvk:SK 820 rvk:SK 980
2	Probability and Heat Kernel Estimates for Lévy(-Type) Processes Kühn, 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. Wahrscheinlichkeitstheorie Lévy-typ Prozess Feller Prozess Lévy Prozess Parametrix-Konstruktion Momente stochastische Differentialgleichungen Existenz stable-like probability theory Lévy-type processes Feller processes Lévy processes parametrix construction stochastic differential equation moments existence result stable-like heat kernel estimates ddc:510 rvk:SK 820 Wahrscheinlichkeitstheorie
3	Lévy-Type Processes under Uncertainty and Related Nonlocal Equations Hollender, Julian 12 October 2016 (has links) The theoretical study of nonlinear expectations is the focus of attention for applications in a variety of different fields — often with the objective to model systems under incomplete information. Especially in mathematical finance, advances in the theory of sublinear expectations (also referred to as coherent risk measures) lay the theoretical foundation for modern approaches to evaluations under the presence of Knightian uncertainty. In this book, we introduce and study a large class of jump-type processes for sublinear expectations, which can be interpreted as Lévy-type processes under uncertainty in their characteristics. Moreover, we establish an existence and uniqueness theory for related nonlinear, nonlocal Hamilton-Jacobi-Bellman equations with non-dominated jump terms. info:eu-repo/classification/ddc/510 ddc:510
4	Probability and Heat Kernel Estimates for Lévy(-Type) Processes Kühn, 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. info:eu-repo/classification/ddc/510 ddc:510 Wahrscheinlichkeitstheorie

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