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Portfolio Optimization and Stochastic Control under Transaction Costs

  • This thesis is concerned with stochastic control problems under transaction costs. In particular, we consider a generalized menu cost problem with partially controlled regime switching, general multidimensional running cost problems and the maximization of long-term growth rates in incomplete markets. The first two problems are considered under a general cost structure that includes a fixed cost component, whereas the latter is analyzed under proportional and Morton-Pliska transaction costs. For the menu cost problem and the running cost problem we provide an equivalent characterization of the value function by means of a generalized version of the Ito-Dynkin formula instead of the more restrictive, traditional approach via the use of quasi-variational inequalities (QVIs). Based on the finite element method and weak solutions of QVIs in suitable Sobolev spaces, the value function is constructed iteratively. In addition to the analytical results, we study a novel application of the menu cost problem in management science. We consider a company that aims to implement an optimal investment and marketing strategy and must decide when to issue a new version of a product and when and how much to invest into marketing. For the long-term growth rate problem we provide a rigorous asymptotic analysis under both proportional and Morton-Pliska transaction costs in a general incomplete market that includes, for instance, the Heston stochastic volatility model and the Kim-Omberg stochastic excess return model as special cases. By means of a dynamic programming approach leading-order optimal strategies are constructed and the leading-order coefficients in the expansions of the long-term growth rates are determined. Moreover, we analyze the asymptotic performance of Morton-Pliska strategies in settings with proportional transaction costs. Finally, pathwise optimality of the constructed strategies is established.

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Author:Yaroslav Melnyk
URN (permanent link):urn:nbn:de:hbz:386-kluedo-40733
Advisor:Ralf Korn
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2015/05/13
Year of Publication:2015
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2015/05/11
Date of the Publication (Server):2015/05/18
Tag:Asymptotic Expansion; Leading-Order Optimality; Pathwise Optimality; Quasi-Variational Inequalities; Regime Shifts; Stochastic Impulse Control; Transaction Costs
Number of page:XI, 169
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Cxx Representations of solutions / 35C20 Asymptotic expansions
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes / 60G40 Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES / 91Gxx Mathematical finance / 91G10 Portfolio theory
91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES / 91Gxx Mathematical finance / 91G80 Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
93-XX SYSTEMS THEORY; CONTROL (For optimal control, see 49-XX) / 93Exx Stochastic systems and control / 93E20 Optimal stochastic control
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 13.02.2015