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The anisotropic p-capacity and the anisotropic Minkowski inequality

Science China Mathematics 2022年第3期65卷 559-582页

作者： Chao Xia Jiabin Yin School of Mathematical Sciences Xiamen UniversityXiamen 361005China

In this paper,we prove a sharp anisotropic L;Minkowski inequality involving the total L^(p)anisotropic mean curvature and the anisotropic p-capacity for any bounded domains with smooth boundary in R^(n).As consequences,we obtain an anisotropic Willmore inequality,a sharp anisotropic Minkowski inequality for outward F-minimising sets and a sharp volumetric anisotropic Minkowski inequality.For the proof,we utilize a nonlinear potential theoretic approach which has been recently developed by Agostiniani et al.(2019).

关键词： Minkowski inequality anisotropic mean curvature anisotropic p-Laplacian nonlinear potential theory p-capacity

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Towards an L^p potential theory for Sub-Markovian Semigroups:Kernels and Capacities

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Acta Mathematica Sinica,English Series 2006年第4期22卷 1227-1250页

作者： Niels JACOB RenéL.SCHILLING Department of Mathematics University of Wales SwanseaSingleton ParkSwansea SA 2 8 PP.United Kinadom FB 12-Mathematik Philipps-Universitat MarburgD-35032 MarburgGermany

We study in fairly general measure spaces （X,μ） the （non-linear） potential theory of L^p sub-Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence （and regularity） of such densities. We give various （dual） representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of R^n where abstract Bessel potential spaces can be identified with concrete function spaces.

关键词： nonlinear potential theory （r,p）-capacity Bessel potential space gamma.transform sub-Markovian semigroup

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Filtration Consistent nonlinear Expectations and Evaluations of Contingent Claims

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Acta Mathematicae Applicatae Sinica 2004年第2期20卷 191-214页

作者： ShigePeng InstituteofMathematics ShandongUniversity250100JinanChina

We will study the following problem.Let X_t,t∈[0,T],be an R^d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before t.Thus at time T,the random value of Y(ω) will become known to this agent.The question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε_t[Y] to define the value of Y given by this agent at time t.Thisoperator ε_t[·] assigns an (X_s)0(?)s(?)T-dependent random variable Y to an (X_s)0(?)s(?)t-dependent random variableε_t[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σ containing an unknown parameter θ=θ_t.We then consider theso called super evaluation when the agent is a seller of the asset Y.We will prove that such super evaluation is afiltration consistent nonlinear expectation.In some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a g-evaluation.We also consider the correspondingnonlinear Markovian situation.

关键词： option pricing measure of risk backward stochastic differential equation nonlinear potential theory nonlinear Markov property dynamic programming principle

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Filtration Consistent nonlinear Expectations and Evaluations of Contingent Claims

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Acta Mathematicae Applicatae Sinica 2004年第2期 15-38页

作者： Shige Peng Institute of Mathematics,Shandong University,250100 Jinan,China

We will study the following problem.Let X,t∈[0,T],be an R~d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before t.Thus at time T,the random value of Y(ω) will become known to this agent.The question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε[Y] to define the value of Y given by this agent at time t.Thisoperator ε[·] assigns an (X)0(?)s(?)T-dependent random variable Y to an (X)0(?)s(?)t-dependent random variableε[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σcontaining an unknown parameter θ=θ.We then consider theso called super evaluation when the agent is a seller of the asset Y.We will prove that such super evaluation is afiltration consistent nonlinear expectation.In some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a g-evaluation.We also consider the correspondingnonlinear Markovian situation.

关键词： option pricing measure of risk backward stochastic differential equation nonlinear potential theory nonlinear Markov property dynamic programming principle

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