Duality in Optimization Financial Problems
The first part of the monograph deals with the more classical concepts and results of duality methods in continuous-time context, developed over the last ten years. The second part, devoted to the semimartingale framework, is more subjective, less standard, and with a special emphasis put on recently developed models. The outline of the monograph is organized as follows. Chapter 1 solves the problem maximizing the expected utility of consumption over the planning horizon, or to maximize the expected utility wealth at the end of the planning horizon, or to maximize some combination of these quantities. The agent under consideration is a small investor in the sense that his actions does not influence to market prices. The market is assumed to be a continuous-time and complete. In Chapter 2, the stochastic control problem of maximizing expected utility from terminal wealth and consumption process, when the portfolio is constrained, is studied We still keep the existing setting, that is, the asset prices follows continuous-time, Ito processes. General existence results are established for optimal consumption plans, by suitable embedding the constrained problem in an appropriate family of unconstrained ones, and finding a member of this family for which the corresponding optimal policy obeys the constraints. Chapter 3 presents a unified, simple and very effective "trick" to derive the dual form of the original problem, which works in a wide range of cases. Explicit results for some] well-known problems are provided, and it is shown how to use this method in order to derive more simply results of the considered examples. Chapter 4 studies the problem of maximizing expected utility of terminal wealth in the framework of a general incomplete semimartingale model of a financial market. The necessary and sufficient condition on a utility function for the existence the optimal solution is, either the requirement that the asymptotic elasticity of the utility function is strictly less then, or the value function of the dual problem is finite. Chapter 5 analyzes the stochastic optimization problem under constraints in a general framework, including financial models with constrained portfolios, labor income and large investor models. We also impose American type constraint on the state space process. General objective functions including deterministic or random utility functions and shortfall risk loss functions, as well as the utility function of a consumption rate process, are considered. We first prove existence and uniqueness result to this optimization problem. In a second part, we develop a dual formulation under minimal assumptions on the objective functions (eventually also establish a new probability space), which are the analogue of the asymptotic elasticity. Chapter 6 analyzes the same problem described in Chapter 5, however, with state processes of a different stochastic structure. Here, the convex constraints are formulated not in terms of the integrands but in terms of its proportions. From the viewpoint of finance, it means that the convex constraints are not imposed on the amount or share of risky invested assets but on the proportions of portfolio. In this chapter, the general optional decomposition under constraints in multiplicative form is employed in place of its version in additive form, which is used in Chapter 5. Concluding Remarks are offered in the last chapter. In Appendix, we introduce some important results, which are frequently used throughout the monograph, like the properties of the asymptotic elasticity, the optional decomposition under constraints (both in additive and multiplicative form). For completeness, we also prove a stochastic control lemma of the optional decomposition under multiplicative form, which was given without proof in Follmer and Kramkov (1997). (fragment of text)
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