Stochastic impulse control and asset allocation with liquidity breakdowns [Elektronische Ressource] / Peter M. Diesinger

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• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2009
• Nombre de lectures: 11
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Stochastic Impulse Control
and
Asset Allocation with Liquidity Breakdowns
Peter M. Diesinger
January 20, 2009
Supervised by
Prof. Dr. Ralf Korn
Datum der Disputation: 14. Mai 2009
Vom Fachbereich Mathematik der Technischen Universit˜at Kaiserslautern
zur Verleihung des akademischen Grades Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation.
1. Gutachter: Prof. Dr. Ralf Korn 2. Gutachterin: Prof. Dr. Nicole B˜auerle
D 386Stochastic Impulse Control
and
Asset Allocation with Liquidity Breakdowns
Peter M. Diesinger
January 20, 2009
Supervised by
Prof. Dr. Ralf Korn
Datum der Disputation: 14. Mai 2009
Vom Fachbereich Mathematik der Technischen Universit˜at Kaiserslautern
zur Verleihung des akademischen Grades Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation.
1. Gutachter: Prof. Dr. Ralf Korn 2. Gutachterin: Prof. Dr. Nicole B˜auerle
D 386Stochastic Control and Financial Mathematics Group
Department of Mathematics
University of Kaiserslautern
˜Erwin-Schrodinger-Stra…e
67663 Kaiserslautern
e-mail: [e-mail] thanks to Ralf Korn for his excellent mentoring over the past few years. He always
found the time to help me with his good advice and constructive criticism. Many thanks
toHolgerKraftforthegoodcollaboration. Thesecondchapterofthisthesisisbasedonan
idea of his; parts of this chapter will be published in Finance and Stochastics and are the
result of the cooperation with Holger Kraft and Frank Seifried. I thank Frank Seifried for
many valuable discussions. Frank is always a good ally to tackle all kinds of mathematical
problems. Thanks to the other colleagues in the Financial Mathematics and Stochastic
Control group for the pleasant working atmosphere, in particular to Stefanie Muller˜ and
J˜orn Sass. I would like to thank Nicole B˜auerle for refereeing this thesis. My thanks to
my friends and family for their constant support. Special, heartfelt thanks to my parents
Aloys and Margrit Diesinger.Introduction
Continuous stochastic control theory has found many applications in optimal investment.
However, it lacks some reality, as it is based on the assumption that interventions are
costless, whichyieldsoptimalstrategieswherethecontrollerhastointerveneateverytime
instant. Thisthesisconsistsoftheexaminationoftwotypesofmorerealisticcontrolmeth-
ods with possible applications.
In the flrst chapter, we study the stochastic impulse control of a difiusion process. We
suppose that the controller minimizes expected discounted costs accumulating as running
andcontrollingcost,respectively. Eachcontrolactioncausescostswhichareboundedfrom
below by some positive constant. This makes a continuous control impossible as it would
lead to an immediate ruin of the controller. In comparison to continuous control, apart
from the pioneering work by Bensoussan and Lions [3] and [2], Menaldi [19], Richard [25],
and Harrison, Selke and A. Taylor [11], there is only very few literature on this problem.
The objective of the flrst part of Chapter 1 is to give a rigorous development of the rele-
vant theory, where our guideline is to establish veriflcation and convergence results under
minimal assumptions, without focusing on the existence of solutions to the corresponding
(quasi-)variational inequalities. If the impulse control problem can be characterized or
approximated by (quasi-)variational inequalities, it remains to solve these equations. For
many problems, such as applications in portfolio selection for stock markets, an impulse
controlapproachistheappropriatemodel. However,itisverydi–culttoobtainexplicitan-
alytic solutions. Papers dealing with applications of stochastic impulse control to flnancial
market models include Buckley and Korn [5], Eastham and Hastings [8], Jeanblanc-Picqu¶e
[14], Korn [16], and Morton and Pliska [23]. In Section 1.2, we solve the stochastic im-
pulse control problem for a one-dimensional difiusion process with constant coe–cients
and convex running costs. Further, in Section 1.3, we solve a particular multi-dimensional
example, where the uncontrolled process is given by an at least two-dimensional Brownian
motion and the cost functions are rotationally symmetric. By symmetry, this problem
can be reduced to a one-dimensional problem. In the last section of the flrst chapter, we
suggestanewimpulsecontrolproblem,wherethecontrollerisinadditionallowedtoinvest
his initial capital into a market consisting of a money market account and a risky asset.
Trading in this market involves transaction costs. The costs which arise upon controlling
thedifiusionprocessandupontradinginthismarkethavetobepaidoutofthecontroller’s
bond holdings. The aim of the controller is to minimize the running costs, caused by the
abstract difiusion process, without getting ruined. This combines the general theory of
stochastic impulse control with the particular case of optimal investment in a market with
transaction costs. The linkage arises by the restriction of the set of admissible strategies.
Asopposedtopapersdealingwiththeextensionofthestandardmarketmodelbyincluding
transactioncosts,thereisanotherstrandofliteratureextendingthestandardmarketmodel
by taking liquidity constraints into account. For instance, Longstafi [20] considers the
portfolio problem of an investor who can only implement portfolio strategies with flnitevariation. Schwartz and Tebaldi [27] assume that an investor cannot trade a risky asset at
all, i.e. the trading interruption is permanent. Rogers [26] analyzes the portfolio decision
of an investor who is constrained to change his strategy at discrete points in time only,
althoughtradingtakesplacecontinuously. Kahl, LiuandLongstafi[15], andLongstafi[21]
consideraninvestmentproblemwheretheadventofatradinginterruptionisknown. These
papersarerelatedtothesecondmainaspectofthisthesispresentedinthesecondchapter.
There, we suggest a new model for illiquidity. This chapter is based on a paper which is
jointworkwithHolgerKraftandFrankSeifried[7]. Weanalyzetheportfoliodecisionofan
investor trading in a market where the economy switches randomly between two possible
states,anormalstatewheretradingtakesplacecontinuously,andanilliquiditystatewhere
trading is not allowed at all. We allow for jumps in the market prices at the beginning
and at the end of a trading interruption. Section 2.1 provides an explicit representation of
the investor’s portfolio dynamics in the illiquidity state in an abstract market consisting
of two assets. In Section 2.2 we specify this market model and assume that the investor
maximizes expected utility from terminal wealth. We establish convergence results, if the
maximal number of liquidity breakdowns goes to inflnity. In the Markovian framework of
Section 2.3, we provide the corresponding Hamilton-Jacobi-Bellman equations and prove a
veriflcation result. We apply these results to study the portfolio problem for a logarithmic
investor and an investor with a power utility function, respectively. Further, we extend
this model to an economy with three regimes. For instance, the third state could model
an additional flnancial crisis where trading is still possible, but the excess return is lower
and the volatility is higher than in the normal state.