Application of conditional set theory to stochastic optimization
Final Report Abstract
Conditional set theory is an approach to local structures which arise, for instance, in dynamic and robust settings. A conditional set encodes in a consistent way an information flow such that the resulting conditional set operations systematically preserve measurability. We introduced the concept of a conditional metric space and show how conditionally semi-continuous stable functions extend the notion of normal integrands. By means of conditional compactness we derived existence of parameter-dependent control problems and fully coupled forward-backward systems on general conditional metric spaces. The representation of nonlinear expectation was based on convex duality arguments. We provided a Fenchel-Moreau theorem for vector-valued convex functions. To that end, we provided a general extension procedure which allows to extend classical functions to stable functions in conditional spaces, which makes the tool from conditional analysis applicable, in particular by means of the conditional Fenchel-Moreau theorem. As a by-product we derived a canonical identification of strongly measurable functions with stable functions in conditional set theory. For the representation of nonlinear expectations we further showed a conditional version of Choquet’s capacitability theorem and provided representation results by means of nonlinear semigroups (so-called Nisio-semigroups). We developed a conditional measure theory which for instance includes the theorems of Fubini, Radon-Nikodým, Daniell-Stone or the Riesz-representation in a conditional setting. A main result showed how the desintegration theorem for conditional expectations extends from standard Borel-spaces to general measure spaces by replacing kernels with conditional measures. Several recent works suggest that every classical result can be translated and formulated in conditional set theory. It was shown a transfer principle based on second-order arithmetic, which suffices for the bulk of classical mathematics, including real analysis and measure theory. Finally, we studied several robust control problems in finance such as robust utility maximization, model-free super-hedging or risk-based pricing when the underlying set of possible financial market models is not dominated.
Publications
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(2018) Measures and Integrals in Conditional Set Theory. Set-Valued Var. Anal (Set-Valued and Variational Analysis) 26 (4) 947–973
Jamneshan, Asgar; Kupper, Michael; Streckfuß, Martin
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(2019) Exponential utility maximization under model uncertainty for unbounded endowments. Ann. Appl. Probab. (The Annals of Applied Probability) 29 (1) 577-612
Bartl, Daniel
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(2017) A Fenchel-Moreau theorem for L¯0-valued functions. Journal of Convex Analysis 26 (2) 593-603
Drapeau, Samuel; Jamneshan, Asgar; Kupper, Michael
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Multidimensional quadratic BSDEs with separated generators. Electronic Communications in Probability, 22, 1–10, 2017
A. Jamneshan, M. Kupper, and P. Luo
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(2018) A transfer principle for second-order arithmetic, and applications. Journal of Logic and Analysis, 10 (8) 1-32
Carl, Merlin; Jamneshan, Asgar
