Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The first approach initiated by Merton [Mer69, Mer71] to solve utility maximization portfolio
problems in continuous time is based on stochastic control theory. The idea of Merton
was to interpret the maximization portfolio problem as a stochastic control problem where
the trading strategies are considered as a control process and the portfolio wealth as the
controlled process. Merton derived the Hamilton-Jacobi-Bellman (HJB) equation and for
the special case of power, logarithm and exponential utility functions he produced a closedform
solution. A principal disadvantage of this approach is the requirement of the Markov
property for the stocks prices. The so-called martingale method represents the second
approach for solving utility maximization portfolio problems in continuous time. It was
introduced by Pliska [Pli86], Cox and Huang [CH89, CH91] and Karatzas et al. [KLS87]
in di erent variant. It is constructed upon convex duality arguments and allows one to
transform the initial dynamic portfolio optimization problem into a static one and to resolve
it without requiring any \Markov" assumption. A de nitive answer (necessary and
su cient conditions) to the utility maximization portfolio problem for terminal wealth has
been obtained by Kramkov and Schachermayer [KS99]. In this thesis, we study the convex
duality approach to the expected utility maximization problem (from terminal wealth) in
continuous time stochastic markets, which as already mentioned above can be traced back
to the seminal work by Merton [Mer69, Mer71]. Before we detail the structure of our
thesis, we would like to emphasize that the starting point of our work is based on Chapter
7 in Pham [P09] a recent textbook. However, as the careful reader will notice, we have
deepened and added important notions and results (such as the study of the upper (lower)
hedge, the characterization of the essential supremum of all the possible prices, compare
Theorem 7.2.2 in Pham [P09] with our stated Theorem 2.4.9, the dynamic programming
equation 2.31, the superhedging theorem 2.6.1...) and we have made a considerable e ort
in the proofs. Indeed, several proofs of theorems in Pham [P09] have serious gaps (not to
mention typos) and even
aws (for example see the proof of Proposition 7.3.2 in Pham [P09] and our proof of Proposition 3.4.8). In the rst chapter, we state the expected utility
maximization problem and motivate the convex dual approach following an illustrative
example by Rogers [KR07, R03]. We also brie
y review the von Neumann - Morgenstern
Expected Utility Theory. In the second chapter, we begin by formulating the superreplication
problem as introduced by El Karoui and Quenez [KQ95]. The fundamental result in
the literature on super-hedging is the dual characterization of the set of all initial endowments
leading to a super-hedge of a European contingent claim. El Karoui and Quenez
[KQ95] rst proved the superhedging theorem 2.6.1 in an It^o di usion setting and Delbaen
and Schachermayer [DS95, DS98] generalized it to, respectively, a locally bounded
and unbounded semimartingale model, using a Hahn-Banach separation argument. The
superreplication problem inspired a very nice result, called the optional decomposition
theorem for supermartingales 2.4.1, in stochastic analysis theory. This important theorem
introduced by El Karoui and Quenez [KQ95], and extended in full generality by Kramkov
[Kra96] is stated in Section 2.4 and proved at the end of Section 2.7. The third chapter
forms the theoretical core of this thesis and it contains the statement and detailed
proof of the famous Kramkov-Schachermayer Theorem that addresses the duality of utility
maximization portfolio problems. Firstly, we show in Lemma 3.2.1 how to transform the
dynamic utility maximization problem into a static maximization problem. This is done
thanks to the dual representation of the set of European contingent claims, which can be
dominated (or super-hedged) almost surely from an initial endowment x and an admissible
self- nancing portfolio strategy given in Corollary 2.5 and obtained as a consequence of
the optional decomposition of supermartingale. Secondly, under some assumptions on the
utility function, the existence and uniqueness of the solution to the static problem is given
in Theorem 3.2.3. Because the solution of the static problem is not easy to nd, we will
look at it in its dual form. We therefore synthesize the dual problem from the primal
problem using convex conjugate functions. Before we state the Kramkov-Schachermayer
Theorem 3.4.1, we present the Inada Condition and the Asymptotic Elasticity Condition
for Utility functions. For the sake of clarity, we divide the long and technical proof of
Kramkov-Schachermayer Theorem 3.4.1 into several lemmas and propositions of independent
interest, where the required assumptions are clearly indicate for each step of the
proof. The key argument in the proof of Kramkov-Schachermayer Theorem is an in nitedimensional
version of the minimax theorem (the classical method of nding a saddlepoint
for the Lagrangian is not enough in our situation), which is central in the theory of Lagrange multipliers. For this, we have stated and proved the technical Lemmata 3.4.5 and
3.4.6. The main steps in the proof of the the Kramkov-Schachermayer Theorem 3.4.1 are:
We show in Proposition 3.4.9 that the solution to the dual problem exists and we
characterize it in Proposition 3.4.12.
From the construction of the dual problem, we nd a set of necessary and su cient
conditions (3.1.1), (3.1.2), (3.3.1) and (3.3.7) for the primal and dual problems to
each have a solution.
Using these conditions, we can show the existence of the solution to the given problem
and characterize it in terms of the market parameters and the solution to the dual
problem.
In the last chapter we will present and study concrete examples of the utility maximization
portfolio problem in speci c markets. First, we consider the complete markets case, where
closed-form solutions are easily obtained. The detailed solution to the classical Merton
problem with power utility function is provided. Lastly, we deal with incomplete markets
under It^o processes and the Brownian ltration framework. The solution to the logarithmic
utility function as well as to the power utility function is presented. / AFRIKAANSE OPSOMMING: Die eerste benadering, begin deur Merton [Mer69, Mer71], om nutsmaksimering portefeulje
probleme op te los in kontinue tyd is gebaseer op stogastiese beheerteorie. Merton
se idee is om die maksimering portefeulje probleem te interpreteer as 'n stogastiese
beheer probleem waar die handelstrategi e as 'n beheer-proses beskou word en die portefeulje
waarde as die gereguleerde proses. Merton het die Hamilton-Jacobi-Bellman (HJB)
vergelyking afgelei en vir die spesiale geval van die mags, logaritmies en eksponensi ele
nutsfunksies het hy 'n oplossing in geslote-vorm gevind. 'n Groot nadeel van hierdie benadering
is die vereiste van die Markov eienskap vir die aandele pryse. Die sogenaamde
martingale metode verteenwoordig die tweede benadering vir die oplossing van nutsmaksimering
portefeulje probleme in kontinue tyd. Dit was voorgestel deur Pliska [Pli86], Cox
en Huang [CH89, CH91] en Karatzas et al. [KLS87] in verskillende wisselvorme. Dit word
aangevoer deur argumente van konvekse dualiteit, waar dit in staat stel om die aanvanklike
dinamiese portefeulje optimalisering probleem te omvorm na 'n statiese een en dit op te
los sonder dat' n \Markov" aanname gemaak hoef te word. 'n Bepalende antwoord (met
die nodige en voldoende voorwaardes) tot die nutsmaksimering portefeulje probleem vir
terminale vermo e is verkry deur Kramkov en Schachermayer [KS99]. In hierdie proefskrif
bestudeer ons die konveks dualiteit benadering tot die verwagte nuts maksimering probleem
(van terminale vermo e) in kontinue tyd stogastiese markte, wat soos reeds vermeld is
teruggevoer kan word na die seminale werk van Merton [Mer69, Mer71]. Voordat ons die
struktuur van ons tesis uitl^e, wil ons graag beklemtoon dat die beginpunt van ons werk
gebaseer is op Hoofstuk 7 van Pham [P09] se onlangse handboek. Die noukeurige leser
sal egter opmerk, dat ons belangrike begrippe en resultate verdiep en bygelas het (soos
die studie van die boonste (onderste) verskansing, die karakterisering van die noodsaaklike
supremum van alle moontlike pryse, vergelyk Stelling 7.2.2 in Pham [P09] met ons verklaarde
Stelling 2.4.9, die dinamiese programerings vergelyking 2.31, die superverskansing stelling 2.6.1...) en ons het 'n aansienlike inspanning in die bewyse gemaak. Trouens,
verskeie bewyse van stellings in Pham cite (P09) het ernstige gapings (nie te praat van
setfoute nie) en selfs foute (kyk byvoorbeeld die bewys van Stelling 7.3.2 in Pham [P09]
en ons bewys van Stelling 3.4.8). In die eerste hoofstuk, sit ons die verwagte nutsmaksimering
probleem uit een en motiveer ons die konveks duaale benadering gebaseer op 'n
voorbeeld van Rogers [KR07, R03]. Ons gee ook 'n kort oorsig van die von Neumann -
Morgenstern Verwagte Nutsteorie. In die tweede hoofstuk, begin ons met die formulering
van die superreplikasie probleem soos voorgestel deur El Karoui en Quenez [KQ95]. Die
fundamentele resultaat in die literatuur oor super-verskansing is die duaale karakterisering
van die versameling van alle eerste skenkings wat lei tot 'n super-verskans van' n Europese
voorwaardelike eis. El Karoui en Quenez [KQ95] het eers die super-verskansing stelling
2.6.1 bewys in 'n It^o di usie raamwerk en Delbaen en Schachermayer [DS95, DS98] het
dit veralgemeen na, onderskeidelik, 'n plaaslik begrensde en onbegrensde semimartingale
model, met 'n Hahn-Banach skeidings argument. Die superreplikasie probleem het 'n prag
resultaat ge nspireer, genaamd die opsionele ontbinding stelling vir supermartingales 2.4.1
in stogastiese ontledings teorie. Hierdie belangrike stelling wat deur El Karoui en Quenez
[KQ95] voorgestel is en tot volle veralgemening uitgebrei is deur Kramkov [Kra96] is uiteengesit
in Afdeling 2.4 en bewys aan die einde van Afdeling 2.7. Die derde hoofstuk vorm
die teoretiese basis van hierdie proefskrif en bevat die verklaring en gedetailleerde bewys
van die beroemde Kramkov-Schachermayer stelling wat die dualiteit van nutsmaksimering
portefeulje probleme adresseer. Eerstens, wys ons in Lemma 3.2.1 hoe om die dinamiese
nutsmaksimering probleem te omskep in 'n statiese maksimerings probleem. Dit kan gedoen
word te danke aan die duaale voorstelling van die versameling Europese voorwaardelike
eise, wat oorheers (of super-verskans) kan word byna seker van 'n aanvanklike skenking x en
'n toelaatbare self- nansierings portefeulje strategie wat in Gevolgtrekking 2.5 gegee word
en verkry is as gevolg van die opsionele ontbinding van supermartingale. In die tweede plek,
met sekere aannames oor die nutsfunksie, is die bestaan en uniekheid van die oplossing van
die statiese probleem gegee in Stelling 3.2.3. Omdat die oplossing van die statiese probleem
nie maklik verkrygbaar is nie, sal ons kyk na die duaale vorm. Ons sintetiseer dan die
duale probleem van die prim^ere probleem met konvekse toegevoegde funksies. Voordat ons
die Kramkov-Schachermayer Stelling 3.4.1 beskryf, gee ons die Inada voorwaardes en die
Asimptotiese Elastisiteits Voorwaarde vir Nutsfunksies. Ter wille van duidelikheid, verdeel
ons die lang en tegniese bewys van die Kramkov-Schachermayer Stelling ref in verskeie lemmas en proposisies op, elk van onafhanklike belang waar die nodige aannames duidelik
uiteengesit is vir elke stap van die bewys. Die belangrikste argument in die bewys van die
Kramkov-Schachermayer Stelling is 'n oneindig-dimensionele weergawe van die minimax
stelling (die klassieke metode om 'n saalpunt vir die Lagrange-funksie te bekom is nie genoeg
in die geval nie), wat noodsaaklik is in die teorie van Lagrange-multiplikators. Vir
die, meld en bewys ons die tegniese Lemmata 3.4.5 en 3.4.6. Die belangrikste stappe in
die bewys van die die Kramkov-Schachermayer Stelling 3.4.1 is:
Ons wys in Proposisie 3.4.9 dat die oplossing vir die duale probleem bestaan en ons
karaktiriseer dit in Proposisie 3.4.12.
Uit die konstruksie van die duale probleem vind ons 'n versameling nodige en voldoende
voorwaardes (3.1.1), (3.1.2), (3.3.1) en (3.3.7) wat die prim^ere en duale probleem
oplossings elk moet aan voldoen.
Deur hierdie voorwaardes te gebruik, kan ons die bestaan van die oplossing vir die
gegewe probleem wys en dit karakteriseer in terme van die mark parameters en die
oplossing vir die duale probleem.
In die laaste hoofstuk sal ons konkrete voorbeelde van die nutsmaksimering portefeulje
probleem bestudeer vir spesi eke markte. Ons kyk eers na die volledige markte geval waar
geslote-vorm oplossings maklik verkrygbaar is. Die gedetailleerde oplossing vir die klassieke
Merton probleem met mags nutsfunksie word voorsien. Ten slotte, hanteer ons onvolledige
markte onderhewig aan It^o prosesse en die Brown ltrering raamwerk. Die oplossing vir
die logaritmiese nutsfunksie, sowel as die mags nutsfunksie word aangebied.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/5259 |
| Date | 12 1900 |
| Creators | Tchamga, Nicole Flaure Kouemo |
| Contributors | Ghomrasni, R., University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 98 p. |
| Rights | University of Stellenbosch |
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