Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

Bosi, Gianni; Mehta, Ghanshyam B.

doi:10.1016/s0304-4068(02)00058-7

Cited by 33 publications

(23 citation statements)

References 20 publications

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“…A proof for the lower semicontinuous case was primarily addressed by Rader [1963] Proof. The first part of the proposition is a direct application of the results in [Bosi and Mehta, 2002]. For the second part, let us defineρ = h • ρ for a lower semicontinuous risk measure ρ : X → [−∞, ∞] and a lower semicontinuous increasing function h : Im (ρ) → R. By relation (2.13),…”

Section: Definition 21 (Lower Semicontinuous Risk Orders) a Risk Ormentioning

confidence: 99%

“…To this end, we choose the framework where X is a locally convex topological vector space. It is a nontrivial result by Bosi and Mehta [2002] that lower semicontinuous preference orders admit a lower semicontinuous numerical representation. We use this result to state that any lower semicontinuous risk order can be represented by a lower semicontinuous risk measure.…”

Section: Introductionmentioning

confidence: 99%

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Risk Preferences and Their Robust Representation

2013

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The goal of this thesis is the conceptual study of risk and its quantification via robust representations.In a first part, we consider risk within a context which extends the notion of "measurable uncertainty" introduced by Frank Knight [1921]. Mathematically, the risk perception of risky elements in a convex set X is expressed by a preference order having the properties of quasiconvexity and monotonicity. These properties are the appropriate translation of the two consensual statements that "diversification should not increase the risk" and "the better for sure, the less risky". Such a preference order will be called a risk order. We keep full latitude on the choice of the underlying setting and thus leave room for different interpretations of risk. Typical examples for X are the space of random variables on a given probability space, the convex set of probability distributions on the real line, or the cone of consumption streams. Risk orders can be represented by numerical representations ρ : states a one-to-one correspondence between risk orders, risk measures, and risk acceptance families. Further properties such as convexity, positive homogeneity, or cash-(sub)additivity are then characterised on these three levels.We then study risk orders on a locally convex topological vector space X . Our main theorem states that any lower semicontinuous risk measure ρ has a unique robust representation of the formwhere R :It is actually the leftinverse in the second argument of the minimal penalty functional αmin (x * , m) = sup x∈A m x * , −x . Here, K• is a polar convex cone in the dual space X * . The proof of uniqueness in this natural context of lower semicontinuity is technically involved, and it is new in the general theory of quasiconvex duality. We also prove a robust representation for risk measures on convex set as needed for risk orders on probability distributions or consumptions streams. We finally provide answers to the delicate question, under which circumstances monotonicity alone ensures lower semicontinuity of the risk order.To finish this first part, we specialize our results to various typical settings. In the case of random variables, we explicitly compute the robust representation of canonical examples such as the certainty equivalent, or the economic index of riskiness. We also show that "Value at Risk" is a risk measure on the level of probability distributions and derive its robust representation. For consumption streams, we obtain a robust representation of the intertemporal utility functional of Hindy, Huang and Kreps. For stochastic kernels, we prove a general separation theorem for risk orders which distinguishes between "model risk" and "distributional risk".In the second part of the thesis, we weaken the requirement of completeness of the preferences, that is, the necessity of deciding whether one element is preferable or not to the other. We introduce the concept of a preference order which might require additional information in order to be expressed. In a first section we ii p...

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Section: Definition 21 (Lower Semicontinuous Risk Orders) a Risk Ormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Risk Preferences and Their Robust Representation

2013

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“…Recently, Acharjee and Tripathy [1] used the concept of bitopological space to reveal poverty patterns and equilibria between strategies of consumers and governments. For some other applications, one may refer to the extensive works of Bosi and Mehta [3].…”

Section: Introductionmentioning

confidence: 99%

Note on $p_1$-Lindelof spaces which are not contra second countable spaces in bitopology

Acharjee

Papadopoulos

Tripathy

2018

bspm

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In this article we show that a contra second countable bitopological space is a p1-Lindelöf space, but the converse part is not necessarily true in general. We provide suitable example with the help of concepts of nest and interlocking from other areas related to bitopology. The relation between pairwise regular spaces and p 1 -normal spaces has been investigated. Finally, we propose some open problems which may enrich various concepts related to Lindelöfness in a bitopological space and other areas of mathematical ideas.

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“…have been used for countable representation of utility function. One may refer to Bosi and Mehta [43]; for their interlink of bitopology and choice via utility function.…”

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confidence: 99%

“…The connection between countability and p 1 -Lindelöfness or p 1 -{∅}-generator may help economists to use bitopological space, Lindelöfness etc. in their respective research areas as one may refer to ( [41], [43]), where authors studied utility functions and various results based on compactness, Lindelöfness, order and other properties of bitopology and general topology.…”

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confidence: 99%

$p$-$\mathcal{I}$-generator and $p_1$-$\mathcal{i}$-generator in bitopology

Acharjee

Tripathy

2018

bspm

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In this article we have investigated the relations of $p$-$\mathcal{I}$-generator, $p_1$-$\mathcal{I}$-generator with $p$-Lindel\"{o}f and $p_1$-Lindel\"{o}f using $\tau_i$-codense, $(i,j)$-meager, $(i,j)$-nowhere dense and perfect mapping of bitopological space. The relations between $p$-compactness, $p$-Lindel\"{o}fness, $p_1$-Lindel\"{o}fness and topological ideal, $(i,j)$-meager, $(i,j)$-Baire space in bitopological space are investigated. Some properties are studied on product bitopology using perfect mapping. It can be found that bitopological space has many applications in real life problems. Hence, we hope that this theory will help to fulfill some interlinks which may have applications in near future.

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Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

Cited by 33 publications

References 20 publications

Risk Preferences and Their Robust Representation

Risk Preferences and Their Robust Representation

Note on $p_1$-Lindelof spaces which are not contra second countable spaces in bitopology

$p$-$\mathcal{I}$-generator and $p_1$-$\mathcal{i}$-generator in bitopology

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