Optimal consumption from investment and random endowment in incomplete semimartingale markets

By Ioannis Karatzas, Gordan Zitkovic. 2003

172 citations recorded by the source corpus.

We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of asymptotic elasticity of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumption/terminal wealth problems, in a common framework. To make the duality approach possible, we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of $L^1$ to its topological bidual $(L^{\infty})^*$, a space of finitely-additive measures. As an application, we treat the case of a constrained It\^ o-process market-model.

Subfields: Economics and Econometrics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/f17bdd27b180f104456d65e44efd553e2f7f776c

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Citations in the corpus

Convex compactness and its applications (2007, result): “In addition to existence results for the usual optimization problems in quantitative finance (see [5,6,12,14,15,19,23,24,26]), some recent results dealing with stability properties of those problems testify even stronger to the indispensability of L+ (and the related convergence in probability) in t”
Explicit solutions of some utility maximization problems in incomplete markets (2004, background): “This generalization of the Merton problem has been studied under very weak assumptions on the asset prices; for instance see [1] or [9].”
Finitely Additive Probabilities and the Fundamental Theorem of Asset Pricing (2009, unknown): “The construction of Y Q in [18] is messier than the one provided above, exactly because condition (3) of Definition 2 is not assumed to hold.”
ON UTILITY MAXIMIZATION IN DISCRETE-TIME FINANCIAL MARKET MODELS (2005, background): “One may even consider even random utility functions U(x,ω); see [19] and [1].”
A dual characterization of self-generation and exponential forward performances (2008, background): “An axiomatic treatment of a class of forward performances by Zariphopoulou and Žitković in [38] is based on an implenetation of this idea in the context of the risk-measure theory, but without a fixed finite investment horizon.”
Portfolio choice under dynamic investment performance criteria (2009, result): “If the terminal utility is taken to be state-dependent, VT (x;T ) 2 FT ; (see, for example, [12], [28] as well as [3], [6] and [10]), the traditional and new formulations are, essentially, identical in [0; T ] : We conclude by mentioning that there is a vast literature on the specication, construct”
Optimal investment with random endowments in incomplete markets (2004, background): “As a result, the optimal investment problem becomes equivalent to one without random endowments but with an augmented initial capital [see Karatzas and Shreve (1998), Chapter 4].”
Option Pricing: Valuation Models and Applications (2004, unknown): “This approach to American option pricing can be traced to Kim (1990), Jacka (1991), and Carr et al.”
Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment (2005, methodology): “Karatzas and Žitković (2003) established existence and uniqueness of an optimal consumption process in an incomplete semimartingale market in the presence of a bounded random endowment.”