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Dual Representation of Convex Increasing Functionals with Applications to Finance

Dual Representation of Convex Increasing Functionals with Applications to Finance

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TANGPI NDOUNKEU, Ludovic, 2015. Dual Representation of Convex Increasing Functionals with Applications to Finance [Dissertation]. Konstanz: University of Konstanz

@phdthesis{TangpiNdounkeu2015Repre-31519, title={Dual Representation of Convex Increasing Functionals with Applications to Finance}, year={2015}, author={Tangpi Ndounkeu, Ludovic}, address={Konstanz}, school={Universität Konstanz} }

Dual Representation of Convex Increasing Functionals with Applications to Finance eng Tangpi Ndounkeu, Ludovic 2015-08-03T12:29:13Z terms-of-use 2015 Tangpi Ndounkeu, Ludovic 2015-08-03T12:29:13Z This thesis deals with the dual representation of various nonlinear functionals and provides applications to financial mathematics under model uncertainty.<br /><br />In the first part of the thesis, we begin by assuming that a fixed reference probability measure is given, and we work on a Brownian filtered probability space (Ω, F , (F<sub>t</sub>)<sub>t≥0</sub>, P ). In this setting, our study of dual representation focuses on minimal supersolutions of back- ward stochastic differential equations (BSDEs) with convex generators. These are convex increasing functionals on a space of non-bounded, but integrable random variables. We derive a dual representation under weak requirements on the generator of the equation. On the other hand, we show that any dynamic risk measure satisfying such a representation stems from a BSDE. As an application, we study the utility maximization problem of an agent with non-zero endowment, and whose preferences are modeled by the maximal subsolution of a BSDE. We prove existence of an optimal trading strategy and relate our existence result to the existence of a maximal subsolution to a controlled decoupled FB-SDE. Using BSDE duality, we show that the utility maximization problem can be seen as a robust control problem admitting a saddle point if the generator of the BSDE additionally satisfies a quadratic growth condition. It is then shown that any saddle point of the robust control problem agrees with a primal and a dual optimizer of the utility maximization problem, and can be characterized in terms of the solution of a BSDE.<br /><br />In the second part of the thesis, we drop the assumption of existence of a reference measure, and work on a topological space Ω which is not assumed to be compact. We give two sorts of conditions guaranteeing the dual representation of convex increasing functionals defined on a space of random variables with respect to countably additive measures. The first conditions, which can be viewed as sequential upper semicontinuity assumptions ensure a max-representation on a Stone vector lattice of continuous random variables. The second condition, which can be viewed as sequential lower semicontinuity assumptions yield a sup-representation on the set of bounded upper semicontinuous random variables; and we characterize functionals admitting a representation on the space of bounded measurable random variables. As applications, we derive a version of the fundamental theorem of asset pricing in continuous and discrete time, and for a market allowing static investments in infinitely many options. We introduce a market efficiency condition stronger than "No Free Lunch With Vanishing Risk" which ensures existence of martingale or local martingale measures for continuous or even càdlàg price processes. On the other hand, we allow trading only in the so-called simple strategies.

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