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Josef Anton Strini On Stochastic Optimization Problems and an Application in Finance BestMasters Mit „BestMasters“ zeichnet Springer die besten Masterarbeiten aus, die an renom mierten Hochschulen in Deutschland, Österreich und der Schweiz entstanden sind. Die mit Höchstnote ausgezeichneten Arbeiten wurden durch Gutachter zur Veröf fentlichung empfohlen und behandeln aktuelle Themen aus unterschiedlichen Fachgebieten der Naturwissenschaften, Psychologie, Technik und Wirtschaftswis senschaften. Die Reihe wendet sich an Praktiker und Wissenschaftler gleicherma ßen und soll insbesondere auch Nachwuchswissenschaftlern Orientierung geben. Springer awards “BestMasters” to the best master’s theses which have been com pleted at renowned Universities in Germany, Austria, and Switzerland. The studies received highest marks and were recommended for publication by supervisors. They address current issues from various fields of research in natural sciences, psychology, technology, and economics. The series addresses practitioners as well as scientists and, in particular, offers guidance for early stage researchers. More information about this series at http://www.springer.com/series/13198 Josef Anton Strini On Stochastic Optimization Problems and an Application in Finance Josef Anton Strini Graz, Austria ISSN 26253577 ISSN 26253615 (electronic) BestMasters ISBN 9783658256906 ISBN 9783658256913 (eBook) https://doi.org/10.1007/9783658256913 Library of Congress Control Number: 2019934745 Springer Spektrum © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Spektrum imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature The registered company address is: AbrahamLincolnStr. 46, 65189 Wiesbaden, Germany Acknowledgements First and foremost I would like to express my sincere gratitude to my su- pervisor Priv.-Doz. Dipl.-Ing. Dr.techn. Thonhauser for his remarkable guidance and his continuous support. Not only during the writing of this thesis, also throughout the entire master’s programme his door was always open and he had advice ready, if needed. I am thankful to my fellow students for the teamwork and the mutual support throughout my years of study. Finally, I want to express my profound gratitude to my family, especially to my girlfriend, for their constant encouragement and their unquestioned backing. Josef Anton Strini V Contents 1. Preliminaries 1 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1. Diffusion processes . . . . . . . . . . . . . . . . . . . . 11 1.2.2. Compound Poisson processes . . . . . . . . . . . . . . 21 1.3. Optimal control of Markov processes . . . . . . . . . . . . . . 22 1.3.1. Dynamic programming. . . . . . . . . . . . . . . . . . 24 1.3.2. Verification step . . . . . . . . . . . . . . . . . . . . . 28 2. A singular stochastic control problem 35 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2. Model assumptions . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3. Singular stochastic control theory . . . . . . . . . . . . . . . . 41 2.3.1. The infinite time horizon problem for Markov diffu- sions in Rd . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.2. The singular stochastic control case . . . . . . . . . . 46 3. Establishing the solution 53 3.1. Dynamic programming approach . . . . . . . . . . . . . . . . 55 3.1.1. First step: The Hamilton-Jacobi-Bellman equation . . 56 3.1.2. Second step: HJB solutions exceed the value function 61 3.1.3. Third step: The conjectured optimal policy . . . . . . 72 3.1.4. Fourth step: Optimality of the conjectured solution . 88 3.2. Resulting consequences and outlook . . . . . . . . . . . . . . 93 A. Numerical complement 95 Bibliography 105 VII List of Figures A.1. Second solution iteration . . . . . . . . . . . . . . . . . . . . . 98 A.2. Auxiliary iteration . . . . . . . . . . . . . . . . . . . . . . . . 99 A.3. Convergence of the iteration . . . . . . . . . . . . . . . . . . . 100 A.4. Comparison of the value function and the numerical solution 102 IX 1. Preliminaries 1.1. Introduction This thesis is primarily concerned with a special stochastic optimal control problembasedonthepaper“Capitalsupplyuncertainty,cashholdings,and investment” authored by Julien Hugonnier, Semyon Malamud and Erwan Morellec [6]. In particular for the considered problem, we want to provide the underlying theory about stochastic optimal control, extend and discuss the existing material in the paper mentioned about the theoretical solution and finally solve it numerically. This first chapter contains the basic mathematical theory with which questions and problems in finance and insurance can be handled. In the followingchapterswewillspecifytheconditionsoftheproblemsconsidered in order to use the tools which will be provided here. The starting point buildssomeintroductorytheoryaboutMarkovprocesses, especiallywewill belookingatMarkovdiffusionprocesses,whichplayanimportantroleinthe contextofstochasticoptimalcontrolproblems. Topreparethetheoryabout theseproblemsisthemaingoalwewanttoachieveinthisfirstchapter. The following is mainly based on two books, namely, on the book by Rolski et al. [14] and on the book by Fleming and Soner [5]. In particular on the one hand, regarding Markov processes, we refer to [14, p. 269 - 270 and p. 437 - 443] and [5, p. 125 - 136]. On the other hand concerning stochastic optimal control theory it is referred to [5, p. 136 - 151], which covers this topic in a more general view and to [5, p. 157 - 177], which provides the theory about controlled Markov diffusions in Rn. © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 J. A. Strini, On Stochastic Optimization Problems and an Application in Finance, BestMasters,https://doi.org/10.1007/978-3-658-25691-3_1 2 1.Preliminaries 1.2. Markov processes Starting with Markov processes in discrete time will make it easier to un- derstand how these processes will behave in continuous time. The prob- ability space we are working on is (Ω,F,P). We consider a finite state space Σ such that Σ = {1,2,...,l} and a so called initial distribution α={α ,α ,...,α }∈[0,1]l. α istheprobabilitythattheMarkovprocess 1 2 l i in discrete time (Xn)n∈N0, which is called Markov chain, starts in state i at time 0: P(X =i)=α . 0 i Next it is common that this Markov chain has a possibility to evolve from the initial state to another state and so we set p ∈ [0,1] the probability ij such that X moves in one time step from state i to state j: n P(Xn =j|Xn−1 =i)=pij. Note that p does not depend on n, therefore the considered Markov chain ij is called homogeneous, in contrast to an inhomogeneous one, where the transition probabilities p depend on n. Moreover in this context we call a ij matrix P =(pij)i,j∈Σ a stochastic matrix, if it fulfills (cid:2)l p ≥0 ∀ i,j ∈Σ and p =1,∀ i∈Σ. ij ij j=1 A Markov chain has the special property that its evolution at a time step only depends on the last state which the process has achieved and not on the whole history. The next definition makes this argument precise. Definition 1. A homogeneous Markov chain is a sequence of random vari- ables (Xn)n∈N0 with values in Σ={1,2,...,l}, for wh(cid:3)ich there exists a vec- tor of probabilities α={α ,α ,...,α }∈[0,1]l with l α =1, which is 1 2 l i=1 i called initial distribution and a stochastic matrix P = (pij)i,j∈Σ, which is called one step transition matrix, such that ∀ n∈N and i ,i ,...,i ∈Σ 0 0 1 n P(X =i ,X =i ,...,X =i )=α p ···p . 0 0 1 1 n n i0 i0i1 in−1in For our purposes it is necessary to consider Markov processes in continu- ous time and with continuous state space. Therefore, we introduce some
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