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José Miguel Zapata

Publications and source records attributed to José Miguel Zapata.

10 recordsLinked to original sources
arXiv2025

Maxitive monetary risk measures: worst-case risk assessment and sharp large deviations

In decision making under uncertainty and risk, worst-case risk assessments are often conducted using maxitive monetary risk measures. In this article, we study maxitive monetary risk measures on the space $L^0$ of all random variables identified modulo almost sure equality. We prove that a monetary risk measure is maxitive and continuous from below if and only if it is a penalized maximum loss. Furthermore, we characterize the maximum loss as the unique maxitive and law-invariant monetary risk measure. We apply the results to large deviation theory by providing a general criterion to establish a sharp large deviation estimate for sequences of probability measures. We use these findings to provide a formula for the asymptotics of the distortion-exponential insurance premium principle under risk pooling.

math.PR↗
arXiv2023

Duality and stable compactness in Orlicz-type modules

Orlicz-type modules are module analogues of classical Orlicz spaces. We study duality and stable compactness in Orlicz-type modules. We characterize the conditional Köthe dual of an Orlicz-type module as the space of all $σ$-order continuous module homomorphisms. We find an order continuity criterion for stable compactness in Orlicz-type modules. As an application, we obtain a robust representation result for conditional risk measures on Orlicz spaces.

math.FA↗
arXiv2023

Weakly maxitive set functions and their possibility distributions

The Shilkret integral with respect to a completely maxitive capacity is fully determined by a possibility distribution. In this paper, we introduce a weaker topological form of maxitivity and show that under this assumption the Shilkret integral is still determined by its possibility distribution for functions that are sufficiently regular. Motivated by large deviations theory, we provide a Laplace principle for maxitive integrals and characterize the possibility distribution under certain separation and convexity assumptions. Moreover, we show a maxitive integral representation result for weakly maxitive non-linear expectations. The theoretical results are illustrated by providing large deviations bounds for sequences of capacities, and by deriving a monotone analogue of Cramér's theorem.

math.PR↗
arXiv2022

On the representation of weakly maxitive monetary risk measures and their rate functions

The present paper provides a representation result for monetary risk measures (i.e., monotone translation invariant functionals) satisfying a weak maxitivity property. This result can be understood as a functional analytic generalization of Gärtner-Ellis large deviations theorem. In contrast to the classical Gärtner-Ellis theorem, the rate function is computed on an arbitrary set of continuous real-valued functions rather than the dual space. As an application of the main result, we establish a large deviation result for sequences of sublinear expectations on regular Hausdorff topological spaces.

math.FA↗
arXiv2020

Large deviations built on max-stability

In this paper, we show that the basic results in large deviations theory hold for general monetary risk measures, which satisfy the crucial property of max-stability. A max-stable monetary risk measure fulfills a lattice homomorphism property, and satisfies under a suitable tightness condition the Laplace Principle (LP), that is, admits a dual representation with affine convex conjugate. By replacing asymptotic concentration of probability by concentration of risk, we formulate a Large Deviation Principle (LDP) for max-stable monetary risk measures, and show its equivalence to the LP. In particular, the special case of the asymptotic entropic risk measure corresponds to the classical Varadhan-Bryc equivalence between the LDP and LP. The main results are illustrated by the asymptotic shortfall risk measure.

math.FA↗
arXiv2019

A Boolean valued analysis approach to conditional risk

By means of the techniques of Boolean valued analysis, we provide a transfer principle between duality theory of classical convex risk measures and duality theory of conditional risk measures. Namely, a conditional risk measure can be interpreted as a classical convex risk measure inside of a suitable set-theoretic model. As a consequence, many properties of a conditional risk measure can be interpreted as basic properties of convex risk measures. This amounts to a method to interpret a theorem of dual representation of convex risk measures as a new theorem of dual representation of conditional risk measures. As an instance of application, we establish a general robust representation theorem for conditional risk measures and study different particular cases of it.

math.FA↗
arXiv2018

Parameter-dependent Stochastic Optimal Control in Finite Discrete Time

We prove a general existence result in stochastic optimal control in discrete time where controls take values in conditional metric spaces, and depend on the current state and the information of past decisions through the evolution of a recursively defined forward process. The generality of the problem lies beyond the scope of standard techniques in stochastic control theory such as random sets, normal integrands and measurable selection theory. The main novelty is a formalization in conditional metric space and the use of techniques in conditional analysis. We illustrate the existence result by several examples including wealth-dependent utility maximization under risk constraints with bounded and unbounded wealth-dependent control sets, utility maximization with a measurable dimension, and dynamic risk sharing. Finally, we discuss how conditional analysis relates to random set theory.

math.OC↗
arXiv2018

Boolean-valued models as a foundation for locally $L^0$-convex analysis and Conditional set theory

Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of multi-period mathematical finance. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. By means of Boolean-valued models and its transfer principle we show that any known result on locally convex spaces has a transcription in the frame of locally $L^0$-convex modules which is also true, and that the formulation in conditional set theory of any theorem of classical set theory is also a theorem. We propose Boolean-valued analysis as an analytic framework for the study of multi-period problems in mathematical finance.

math.FA↗
arXiv2017

A study of stability in locally $L^0$-convex modules and a conditional version of James' compactness theorem

Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. In this paper we study locally $L^0$-convex modules, and find exactly which subclass of locally $L^0$-convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory. Second, we provide a version of the classical James' theorem of characterization of weak compactness for conditional Banach spaces. Finally, we state a conditional version of the Fatou and Lebesgue properties for conditional convex risk measures and, as application of the developed theory, we stablish a version of the so-called Jouini-Schachermayer-Touzi theorem for robust representation of conditional convex risk measures defined on a $L^\infty$-type module.

math.FA↗