Portfolio optimization problems : a martingale and a convex duality approach

[摘要] ENGLISH ABSTRACT: Thefirst approach initiated by Merton [Mer69, Mer71] to solve utility maximization portfolioproblems in continuous time is based on stochastic control theory. The idea of Mertonwas to interpret the maximization portfolio problem as a stochastic control problem wherethe trading strategies are considered as a control process and the portfolio wealth as thecontrolled process. Merton derived the Hamilton-Jacobi-Bellman (HJB) equation and forthe special case of power, logarithm and exponential utility functions he produced a closedformsolution. A principal disadvantage of this approach is the requirement of the Markovproperty for the stocks prices. The so-called martingale method represents the secondapproach for solving utility maximization portfolio problems in continuous time. It wasintroduced 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 totransform the initial dynamic portfolio optimization problem into a static one and to resolveit without requiring any \Markov assumption. A de nitive answer (necessary andsu cient conditions) to the utility maximization portfolio problem for terminal wealth hasbeen obtained by Kramkov and Schachermayer [KS99]. In this thesis, we study the convexduality approach to the expected utility maximization problem (from terminal wealth) incontinuous time stochastic markets, which as already mentioned above can be traced backto the seminal work by Merton [Mer69, Mer71]. Before we detail the structure of ourthesis, we would like to emphasize that the starting point of our work is based on Chapter7 in Pham [P09] a recent textbook. However, as the careful reader will notice, we havedeepened 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, compareTheorem 7.2.2 in Pham [P09] with our stated Theorem 2.4.9, the dynamic programmingequation 2.31, the superhedging theorem 2.6.1...) and we have made a considerable e ortin the proofs. Indeed, several proofs of theorems in Pham [P09] have serious gaps (not tomention 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 therst chapter, we state the expected utilitymaximization problem and motivate the convex dual approach following an illustrativeexample by Rogers [KR07, R03]. We also briey review the von Neumann - MorgensternExpected Utility Theory. In the second chapter, we begin by formulating the superreplicationproblem as introduced by El Karoui and Quenez [KQ95]. The fundamental result inthe literature on super-hedging is the dual characterization of the set of all initial endowmentsleading 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 Delbaenand Schachermayer [DS95, DS98] generalized it to, respectively, a locally boundedand unbounded semimartingale model, using a Hahn-Banach separation argument. Thesuperreplication problem inspired a very nice result, called the optional decompositiontheorem for supermartingales 2.4.1, in stochastic analysis theory. This important theoremintroduced 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 chapterforms the theoretical core of this thesis and it contains the statement and detailedproof of the famous Kramkov-Schachermayer Theorem that addresses the duality of utilitymaximization portfolio problems. Firstly, we show in Lemma 3.2.1 how to transform thedynamic utility maximization problem into a static maximization problem. This is donethanks to the dual representation of the set of European contingent claims, which can bedominated (or super-hedged) almost surely from an initial endowment x and an admissibleself- nancing portfolio strategy given in Corollary 2.5 and obtained as a consequence ofthe optional decomposition of supermartingale. Secondly, under some assumptions on theutility function, the existence and uniqueness of the solution to the static problem is givenin Theorem 3.2.3. Because the solution of the static problem is not easy tond, we willlook at it in its dual form. We therefore synthesize the dual problem from the primalproblem using convex conjugate functions. Before we state the Kramkov-SchachermayerTheorem 3.4.1, we present the Inada Condition and the Asymptotic Elasticity Conditionfor Utility functions. For the sake of clarity, we divide the long and technical proof ofKramkov-Schachermayer Theorem 3.4.1 into several lemmas and propositions of independentinterest, where the required assumptions are clearly indicate for each step of theproof. The key argument in the proof of Kramkov-Schachermayer Theorem is an in nitedimensionalversion of the minimax theorem (the classical method ofnding a saddlepointfor 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 and3.4.6. 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 wecharacterize it in Proposition 3.4.12.From the construction of the dual problem, wend a set of necessary and su cientconditions (3.1.1), (3.1.2), (3.3.1) and (3.3.7) for the primal and dual problems toeach have a solution.Using these conditions, we can show the existence of the solution to the given problemand characterize it in terms of the market parameters and the solution to the dualproblem.In the last chapter we will present and study concrete examples of the utility maximizationportfolio problem in speci c markets. First, we consider the complete markets case, whereclosed-form solutions are easily obtained. The detailed solution to the classical Mertonproblem with power utility function is provided. Lastly, we deal with incomplete marketsunder It^o processes and the Brownianltration framework. The solution to the logarithmicutility function as well as to the power utility function is presented.

[发布机构] Stellenbosch University