Annals of Applied Probability

A unified framework for utility maximization problems: An Orlicz space approach

Sara Biagini and Marco Frittelli

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Abstract

We consider a stochastic financial incomplete market where the price processes are described by a vector-valued semimartingale that is possibly nonlocally bounded. We face the classical problem of utility maximization from terminal wealth, with utility functions that are finite-valued over (a, ∞), a∈[−∞, ∞), and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for stochastic integrals that are not necessarily bounded from below. The embedding of the utility maximization problem in Orlicz spaces permits us to formulate the problem in a unified way for both the cases a∈ℝ and a=−∞. By duality methods, we prove the existence of solutions to the primal and dual problems and show that a singular component in the pricing functionals may also occur with utility functions finite on the entire real line.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 3 (2008), 929-966.

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1211819790

Digital Object Identifier
doi:10.1214/07-AAP469

Mathematical Reviews number (MathSciNet)
MR2418234

Zentralblatt MATH identifier
1151.60019

Subjects
Primary: 60G48: Generalizations of martingales 60G44: Martingales with continuous parameter 49N15: Duality theory 91B28
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46N30: Applications in probability theory and statistics 91B16: Utility theory

Keywords
Utility maximization nonlocally bounded semimartingale incomplete market σ-martingale measure Orlicz space convex duality singular functionals

Citation

Biagini, Sara; Frittelli, Marco. A unified framework for utility maximization problems: An Orlicz space approach. Ann. Appl. Probab. 18 (2008), no. 3, 929--966. doi:10.1214/07-AAP469. https://projecteuclid.org/euclid.aoap/1211819790


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