Robust consumption-investment problems with stochastic coefficients [Elektronische Ressource] / vorgelegt von Christoph Wopperer

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• Mathematics

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2010
• Nombre de lectures: 9
• Langue: English

Extrait

Universitat Ulm
Fakultat fur Mathematik und
Wirtschaftswissenschaften
Robust consumption-investment problems
with stochastic coe cients
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakult at fur Mathematik und Wirtschaftswissenschaften
der Universit at Ulm
vorgelegt von
Christoph Wopperer
aus Erlangen
Ulm, November 2010Amtierender Dekan: Prof. Dr. Werner Kratz
1. Gutachter: Prof. Dr. Ulrich Rieder
2.hter: Prof. Dr. Rudiger Kiesel
Tag der Promotion: 14.02.2011Abstract
In this thesis we consider robust consumption-investment problems in a complete di u-
sion market with stochastic coe cients. We assume that the market price of risk process
is unknown. The investor tries to maximize his expected utility under the worst-case
parameter con guration. To solve robust consumption-investment problems, we derive
a stochastic version of the Bellman-Isaac equations for di erential games from the mar-
tingale optimality principle. A formal connection between a solution of these equations
and the robust optimal value function is established by a veri cation theorem. We are
able to solve the Bellman-Isaac equations for power, exponential and logarithmic utility.
In this way we can characterize a robust optimal consumption-investment strategy and a
worst-case market price of risk process in terms of the solution of a backward stochastic
di erential equation ( Bsde). The solution of this Bsde can be explicitly computed in
case of deterministic coe cients. It is given by the unique solution of a partial di erential
equation for the popular model of coe cients driven by a factor process.
Keywords: optimal consumption, stochastic Bellman-Isaac equations, backward stochas-
tic di erential equations, factor model
Zusammenfassung
In dieser Arbeit analysieren wir robuste Konsum-Investitionsprobleme in einem vollst an-
digen Di usionsmarkt mit stochastischen Koe zienten. Es wird angenommen dass der
Marktpreis des Risikos dem Investor nicht bekannt ist. Dieser versucht seinen erwarteten
Nutzen unter der fur ihn schlechtesten Parameterkon guration zu maximieren. Um ro-
buste Konsum-Investitionsprobleme zuosen,l leiten wir ausgehend von dem Martingalop-
timalit atsprinzip eine stochastische Variante der Bellman-Isaac Gleichungen fur Di eren-
tialspiele her. Ein formeller Zusammenhang zwischen einer L osung dieser Gleichungen und
der robust optimalen Wertfunktion wird durch ein Veri kationstheorem hergestellt. Im
Fall von power, exponentiellem und logarithmischem Nutzen k onnen wir die Bellman-Isaac
Gleichungenosen.l So k onnen wir eine optimale Konsum-Investitionsstrategie und einen
worst-case Marktpreis des Risikos mit Hilfe der L osung einer stochastischen Ruc kw arts-
gleichung charakterisieren. Die L osung dieser Ruc kw artsgleichung kann im Fall von deter-
ministischen Koe zienten explizit angegeben werden. In den in der Literatur popul aren
Faktormodellen ist die L osung der stochastischen Ruc kw artsgleichung durch die eindeutige
L osung einer partiellen Di erentialgleichung gegeben.
Schlagw orter : optimaler Konsum, stochastische Bellman-Isaac Gleichungen, stochastis-
che Ruckw artsgleichungen, FaktormodellContents
1 Introduction 1
1.1 Motivation and formulation of the problem . . . . . . . . . . . . . . . . . . 1
1.2 Outline and contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Maxmin problems, BSDEs and BMO-martingales 9
2.1 Some notions about maxmin problems . . . . . . . . . . . . . . . . . . . . 9
2.2 Basic results about backward stochastic di erential equations . . . . . . . . 14
2.2.1 Backward stochastic di erential equations with Lipschitz generator 15
2.2.2 BSDEs with quadratic growth . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Feynman-Kac theorems . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 BMO-martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Robust consumption-investment problems 25
3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The martingale optimality principle . . . . . . . . . . . . . . . . . . . . . . 29
3.3 The stochastic Bellman-Isaac equations . . . . . . . . . . . . . . . . . . . . 33
3.4 Optimality shown by the comparison theorem . . . . . . . . . . . . . . . . 45
4 Special utility functions 49
4.1 Power utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Exponential utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Logarithmic utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Coe cients driven by a stochastic factor process 77
5.1 Power utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Exponential utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Logarithmic utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Bibliography 89
v1 Introduction
1.1 Motivation and formulation of the problem
The main focus of this thesis lies in nding optimal consumption and investment strategies
for portfolio optimization problems with an additional source of uncertainty.
Historically seen, the rst to address the question \how to invest optimally" in a mathe-
matical way, was Harry Markowitz in his seminal paper [Mar52]. He considered a single-
period problem that already contained all the features inherent in any optimal investment
problem. The problem was essentially a static optimization problem subject to a volatility
constraint. The solution could thus be found by means of quadratic programming.
The next major step was taken by Merton [Mer71] who considered dynamic problems.
Namely, he analyzed a continuous-time market model where stock prices are modeled by a
geometric Brownian motion with an investor who wants to maximize his expected utility
from intermediate consumption and terminal wealth. From a mathematical point of view,
the investment problem could then be solved with the method of dynamic programming.
The investment problems considered by Markowitz and Merton can be subdivided into
three mathematical components. The rst one is the description of a market model. More
precisely, a stochastic model that characterizes the time evolution of the di erent assets
under consideration has to be speci ed. Then, feasible strategies have to be de ned. The
investor can trade continuously or only at discrete times, he might be allowed to consume
parts or all of his wealth or not, and he might face additional practical restrictions for
example with respect to short sales or tradeable asset classes and so on. Finally, the
investor’s objective has to be formulated in terms of a mathematical optimization problem.
Mostly, this is done by using functionals that value the expected utility for a given strategy.
Obviously, all these subproblems require sophisticated modeling. The main focus of this
thesis is a closer analysis of the description of the market model. There are numerous
possible choices for the market model. Not only the investor’s asset universe has to be
limited, also the dynamics of the assets have to be speci ed.
However, classical portfolio optimization problems completely ignore the question where
to get the stochastic model for the behavior of the assets from: they postulate one and
compute the optimal consumption and investment decisions given that this model ad-
equately re ects real dynamics and given that the model parameters are fully known.
While the choice of a model seems natural, the availability of its parameters is a critical
assumption. In reality, model parameters have to be estimated from out-of-sample data in
a step preceding the optimization. It is clear that realistic portfolio optimization also has
to include the choice of the model and the estimation of its parameters. The interaction
between the choice of the model, the estimation of its parameters and the computation
of an optimal strategy is depicted schematically in Figure 1.1.
1Estimation Step Optimization Step
Input consumption-investmentestimation of model
historical data policy that is optimalparameters
according to model
possibly update of model
parameters based on
current observations
setting up parametric model
for the asset price behavior
0 T time
Figure 1.1: Estimation and optimization for portfolio optimization problems
As far as the choice of the model is concerned, the critical issue normally consists in
specifying a model for the returns of the stock prices. Indeed, return behavior is not
entirely understood and estimated returns are subject to large estimation errors. Unfor-
tunately, even small perturbations from a set of reference model parameters might lead
to drastically di erent optimal asset allocations.
In this thesis, we will assume that a model for the behavior of the stocks is chosen but that
its parameters are not exactly known. As an illustration of the di culty of estimating
returns from historical data consider the following example: suppose we observe the prices
of d shares evolving according to
1 d TdS = diag(S )( dt +dW ); S = (s ;:::;s ) ;t t t 0 0 0
where W is a d-dimensional Brownian motion and has full rank d. Here, diag(S ) ist t
1 2 dthe diagonal matrix with entriesS ;S ;:::;S . If is constant but unknown, then (event t t
if the volatility matrix