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Statistics & Risk Modeling

with Applications in Finance and Insurance

Editor-in-Chief: Stelzer, Robert

Cite Score 2018: 0.85

SCImago Journal Rank (SJR) 2018: 0.354
Source Normalized Impact per Paper (SNIP) 2018: 0.604

Mathematical Citation Quotient (MCQ) 2018: 0.36

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Volume 31, Issue 2


Stochastic orderings with respect to a capacity and an application to a financial optimization problem

Miryana Grigorova
  • Corresponding author
  • Laboratoire de Probabilités et Modèles Aléatoires (CNRS-UMR 7599); Université Paris Diderot (Paris 7); 5 rue Thomas Mann; 75013 Paris; France
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Published Online: 2014-05-28 | DOI: https://doi.org/10.1515/strm-2013-1151


By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).

Keywords: Stochastic orderings; increasing convex stochastic dominance; Choquet integral; quantile function with respect to a capacity; stop-loss ordering; Choquet expected utility; distorted capacity; generalized Hardy–Littlewood's inequalities; distortion risk measure; premium principle; ambiguity; non-additive probability

AMS 2010: Primary: 60E15; 91B06; 91B30; Secondary: 91B16; 62P05; 28E10

About the article

Accepted: 2014-01-07

Received: 2013-04-11

Published Online: 2014-05-28

Published in Print: 2014-06-28

Citation Information: Statistics & Risk Modeling, Volume 31, Issue 2, Pages 183–213, ISSN (Online) 2196-7040, ISSN (Print) 2193-1402, DOI: https://doi.org/10.1515/strm-2013-1151.

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