Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

Gao, Niushan; Leung, Denny H.; Munari, Cosimo; Xanthos, Foivos

doi:10.1007/s00780-018-0357-7

Cited by 47 publications

(46 citation statements)

References 33 publications

(49 reference statements)

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“…Clearly, (P u n ) is order bounded, say, P u n ≤ a for all n ≥ 1 and some a ∈ X + . Then it follows from Orlicz spaces have been used in mathematical finance and economics as a general framework of model spaces; see, e.g., [5,7,11,12,15]. We state Theorem 2.9 in this setting.…”

Section: Resultsmentioning

confidence: 93%

Smallest order closed sublattices and option spanning

Gao

Leung

2017

Proc. Amer. Math. Soc.

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Let Y be a sublattice of a vector lattice X. We consider the problem of identifying the smallest order closed sublattice of X containing Y . It is known that the analogy with topological closure fails. Let Y o be the order closure of Y consisting of all order limits of nets of elements from Y . Then Y o need not be order closed. We show that in many cases the smallest order closed sublattice containing Y is in fact the second order closure Y o o. Moreover, if X is a σorder complete Banach lattice, then the condition that Y o is order closed for every sublattice Y characterizes order continuity of the norm of X. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset.This notion that options complete markets, pioneered by Ross, is at the core of modern financial economics ([4]), and has been under extensive exploration.

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Section: Resultsmentioning

confidence: 93%

Smallest order closed sublattices and option spanning

Gao

Leung

2017

Proc. Amer. Math. Soc.

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“…4.58] and [13,Th. 32]; the L p -variant follows from the Orlicz space version proved in [21]. For its validity, it is essential that the probability space is nonatomic.…”

Section: Average Quantiles and The Kusuoka Representationmentioning

confidence: 99%

Convex bodies generated by sublinear expectations of random vectors

Molchanov

Turin

2021

Advances in Applied Mathematics

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We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random vectors in Euclidean space. We identify the dual representation of such convex bodies and describe a construction that serves as a building block for all so defined convex bodies. Sublinear expectations are studied in mathematical finance within the theory of risk measures. In this way, tools from mathematical finance yield a whole variety of new geometric constructions.

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“…Dual representations have been widely investigated in the risk measure literature; see, e.g., [Artzner et al, 1999], [Föllmer and Schied, 2002], [Frittelli and Rosazza Gianin, 2002], [Jouini et al, 2006], and [Cheridito et al, 2017] for the convex case and [Cerreia-Vioglio et al, 2011], [Drapeau and Kupper, 2013], and [Gao et al, 2018] for the quasiconvex case. In this section we establish dual representations of general risk measures under both convexity and quasiconvexity.…”

Section: Dual Representationsmentioning

confidence: 99%

Risk measures beyond frictionless markets

Arduca¹,

Munari²

2021

Preprint

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We develop a general theory of risk measures that determines the optimal amount of capital to raise and invest in a portfolio of reference traded securities in order to meet a pre-specified regulatory requirement. The distinguishing feature of our approach is that we embed portfolio constraints and transaction costs into the securities market. As a consequence, we have to dispense with the property of translation invariance, which plays a key role in the classical theory. We provide a comprehensive analysis of relevant properties such as star shapedness, positive homogeneity, convexity, quasiconvexity, subadditivity, and lower semicontinuity. In addition, we establish dual representations for convex and quasiconvex risk measures. In the convex case, the absence of a special kind of arbitrage opportunities allows to obtain dual representations in terms of pricing rules that respect market bid-ask spreads and assign a strictly positive price to each nonzero position in the regulator's acceptance set.

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Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

Cited by 47 publications

References 33 publications

Smallest order closed sublattices and option spanning

Smallest order closed sublattices and option spanning

Convex bodies generated by sublinear expectations of random vectors

Risk measures beyond frictionless markets

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