Eldorado Collection:http://hdl.handle.net/2003/262024-03-29T02:12:40Z2024-03-29T02:12:40ZNon-uniformly parabolic equations and applications to the random conductance modelBella, PeterSchäffner, Mathiashttp://hdl.handle.net/2003/408532022-04-13T22:12:09Z2021-07-30T00:00:00ZTitle: Non-uniformly parabolic equations and applications to the random conductance model
Authors: Bella, Peter; Schäffner, Mathias
Abstract: We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on Zd. In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.2021-07-30T00:00:00ZAnalyse von Wellenpaketen in der periodischen nichtlinearen Schrödingergleichung durch Approximation mittels Coupled Mode GleichungenWahlers, Lisahttp://hdl.handle.net/2003/381442019-07-18T01:40:48Z2018-01-01T00:00:00ZTitle: Analyse von Wellenpaketen in der periodischen nichtlinearen Schrödingergleichung durch Approximation mittels Coupled Mode Gleichungen
Authors: Wahlers, Lisa
Abstract: Der Kern dieser Arbeit ist die Untersuchung von Wellenpaketen in periodischen Strukturen für unterschiedliche Dimensionen. Als asymptotische Skalierung betrachten wir Wellenpakete aus mehreren tragenden Blochwellen mit unterschiedlichen Gruppengeschwindigkeiten. Dadurch leiten wir ein Amplitudensystem erster Ordnung her, die Coupled Mode Gleichungen.
Im eindimensionalen Fall haben die Coupled Mode Gleichungen eine Familie von lokalisierten Solitärwellen, welche durch die Geschwindigkeit parametrisiert sind. Weil diese Solitärwellen in der spektralen Lücke der Coupled Mode Gleichungen existieren, werden sie Gap Solitone genannt. Wir beschäftigen uns mit der Frage, ob für die Coupled Mode Gleichungen in höheren Dimensionen ebenfalls eine Familie von beweglichen Gap Solitonen existiert.
Für den zweidimensionalen Fall und einen Ansatz aus vier Moden finden wir numerisch stehende Solitärwellen und beweisen anschließend die Existenz von lokalisierten Wellen für die Coupled Mode Gleichungen.
Schließlich führen wir eine rigorose Rechtfertigung der Coupled Mode Gleichungen als asymptotisches Modell für die periodische nichtlineare Schrödingergleichung für den allgemeinen Fall von d Dimensionen durch.2018-01-01T00:00:00ZTraveling wave solutions of reaction-diffusion equations with x-dependent combustion type nonlinearitiesBadke, Svenhttp://hdl.handle.net/2003/349632016-07-05T12:55:24Z2016-03-01T00:00:00ZTitle: Traveling wave solutions of reaction-diffusion equations with x-dependent combustion type nonlinearities
Authors: Badke, Sven
Abstract: We investigate the existence and uniqueness of traveling wave solutions of
the reaction-diffusion equation in periodic heterogeneous media. The reaction-diffusion
equation is considered in nondivergence form with no first order term.
Our traveling wave problem is considered in similar form in [1] by Xin in
the special case that the reaction-term is given by a combustion nonlinearity
ƒ = ƒ(u).
We prove the existence of traveling wave solutions in case of a class of nonlinearities
ƒ = ƒ(x, u), which are a generalization of a combustion nonlinearity.
In particular, ƒ is allowed to depend explicitly on x. In case of an additional
assumption on ƒ, we also prove a monotonicity result and a uniqueness result.
References
[1] X. Xin. Existence and uniqueness of travelling waves in a reaction-diffusion equation with combustion nonlinearity. Indiana Univ. Math. J., 40(3):985–1008, 1991.2016-03-01T00:00:00ZUniqueness and regularity for porous media equations with x-dependent coefficientsKoch, Jan-Christopherhttp://hdl.handle.net/2003/339302015-08-12T20:01:51Z2014-01-01T00:00:00ZTitle: Uniqueness and regularity for porous media equations with x-dependent coefficients
Authors: Koch, Jan-Christopher2014-01-01T00:00:00ZA worst-case optimization approach to impulse perturbed stochastic control with application to financial risk managementMönnig, Laurenzhttp://hdl.handle.net/2003/295742016-05-09T10:35:38Z2012-08-07T00:00:00ZTitle: A worst-case optimization approach to impulse perturbed stochastic control with application to financial risk management
Authors: Mönnig, Laurenz
Abstract: This work presents the main ideas, methods and results of the theory of impulse perturbed
stochastic control as an extension of the classic stochastic control theory. Apart from the introduction
and the motivation of the basic concept, two stochastic optimization problems are
the focus of the investigations. On the one hand we consider a differential game as analogue
of the expected utility maximization problem in the situation with impulse perturbation,
and on the other hand we study an appropriate version of a target problem. By dynamic
optimization principles we characterize the associated value functions by systems of partial
differential equations (PDEs). More precisely, we deal with variational inequalities whose
single inequalities comprise constrained optimization problems, where the corresponding admissibility
sets again are given by the seeked value functions. Using the concept of viscosity
solutions as weak solutions of PDEs, we avoid strong regularity assumptions on the value
functions. To use this concept as sufficient verification method, we additionally have to prove
the uniqueness of the solutions of the PDEs.
As a second major part of this work we apply the presented theory of impulse perturbed
stochastic control in the field of financial risk management where extreme events have to be
taken into account in order to control risks in a reasonable way. Such extreme scenarios are
modelled by impulse controls and the financial decisions are made with respect to the worstcase
scenario. In a first example we discuss portfolio problems as well as pricing problems on
a capital market with crash risk. In particular, we consider the possibility of trading options
and study their in
uence on the investor's performance measured by the expected utility of
terminal wealth. This brings up the question of crash-adjusted option prices and leads to
the introduction of crash insurance. The second application concerns an insurance company
which faces potentially large losses from extreme damages. We propose a dynamic model
where the insurance company controls its risk process by reinsurance in form of proportional
reinsurance and catastrophe reinsurance. Optimal reinsurance strategies are obtained by
maximizing expected utility of the terminal surplus value and by minimizing the required
capital reserves associated to the risk process.2012-08-07T00:00:00ZWaves in heterogeneous media: long time behavior and dispersive modelsLamacz, Agneshttp://hdl.handle.net/2003/290812015-08-12T18:04:25Z2011-09-12T00:00:00ZTitle: Waves in heterogeneous media: long time behavior and dispersive models
Authors: Lamacz, Agnes2011-09-12T00:00:00ZOn the local well-posedness of the Kadomtsev-Petviashvili II equationHadac, Martinhttp://hdl.handle.net/2003/246752015-08-13T01:49:52Z2007-08-27T09:48:25ZTitle: On the local well-posedness of the Kadomtsev-Petviashvili II equation
Authors: Hadac, Martin2007-08-27T09:48:25ZWell-posedness results for dispersive equations with derivative nonlinearitiesHerr, Sebastianhttp://hdl.handle.net/2003/228562015-08-13T00:43:34Z2006-08-28T12:14:46ZTitle: Well-posedness results for dispersive equations with derivative nonlinearities
Authors: Herr, Sebastian2006-08-28T12:14:46ZEine analytische Methode zur Punktereduktion und FlächenrekonstruktionGuias, Adina Aureliahttp://hdl.handle.net/2003/216212015-08-12T16:34:32Z2005-09-27T09:31:45ZTitle: Eine analytische Methode zur Punktereduktion und Flächenrekonstruktion
Authors: Guias, Adina Aurelia2005-09-27T09:31:45Z