Jensen’s inequality for g-convex function under g-expectation

Jia, Guangyan; Péng, Shigē

doi:10.1007/s00440-009-0206-x

Cited by 16 publications

(7 citation statements)

References 24 publications

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“…It follows from Example 10 in Peng (1997) to get (vi), and (vii) is derived from Lemma 36.9 in Peng (1997). Since u ∈ C 2 (R) is increasing and concave, we can get (viii) from Theorem 1 in Jia and Peng (2010).…”

Section: B Proofsmentioning

confidence: 94%

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Dynamic Consistent -Maxim Expected Utility

Beißner

Lin

2015

SSRN Journal

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract We establish a class of fully nonlinear conditional expectations. Similarly to the usage of linear expectations when a probabilistic description of uncertainty is present, we observe analogue quantitative and qualitative properties. The type of nonlinearity captures the agents sentiments of optimism and pessimism in an ambiguous environment. Terms of use: Documents in EconStor mayWe then introduce an expected utility under a nonlinear expectation, and show monotonicity and continuity of utility. Risk aversion is characterized, and the properties of the certainty equivalent are discussed. Finally, we derive an Arrow-Pratt approximation of the static certainty equivalent and investigate the dynamic version via recursive equations.

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Section: B Proofsmentioning

confidence: 94%

“…By an application of Theorem 3.2 in Jia and Peng (2010) to e(t, σ), which is independent of E t [X]), the conditional E-concavity, i.e., u(…”

Section: Proof Of Proposition 2 (I)mentioning

confidence: 99%

Dynamic Consistent -Maxim Expected Utility

Beißner

Lin

2015

SSRN Journal

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“…Remark 4.10. As a matter of fact, this phenomenon is also one of the motivations for us to introduce a new concept: g-convex functions (see [17]). …”

Section: A Remark On Jensen's Inequality For G-expectationmentioning

confidence: 98%

Backward stochastic differential equations with a uniformly continuous generator and related g-expectation

Jia

2010

Stochastic Processes and their Applications

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In this paper, we will study a class of backward stochastic differential equations (BSDEs for short), for which the generator (coefficient) g(t, y, z) is Lipschitz continuous with respect to y and uniformly continuous with respect to z. We establish several properties for such BSDEs, including comparison and converse comparison theorems, a representation theorem for g and a continuous dependence theorem. Then we introduce a new class of g-expectation based on such backward stochastic differential equations, and discuss its properties.

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“…We refer Jia and Peng () for Jensen's inequality of g ‐convex function under one‐dimensional g ‐expectation. g ‐convexity is a new notion of convex functions.…”

Section: Multidimensional G‐expectationsmentioning

confidence: 99%

MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g‐EXPECTATION

2014

Mathematical Finance

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This paper deals with multidimensional dynamic risk measures induced by conditional g-expectations. A notion of multidimensional g-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional gexpectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g-risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g-risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for γ -tolerant g-risk measures, and risk contribution for coherent g-risk measures are investigated. Insurance g-risk measure and other ways to induce g-risk measures are also studied at the end of the paper.

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Jensen’s inequality for g-convex function under g-expectation

Cited by 16 publications

References 24 publications

Dynamic Consistent -Maxim Expected Utility

Dynamic Consistent -Maxim Expected Utility

Backward stochastic differential equations with a uniformly continuous generator and related g-expectation

MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g‐EXPECTATION

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