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Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations

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Acta Mathematicae Applicatae Sinica 2011年第4期27卷 647-678页

作者： rainer.buckdahn Département de Mathématiques Université de Bretagne Occidentale6avenue Victor-le-GorgeuB.P.80929285 Brest cedexFrance Institute for Advanced Study Shangdong UniversityJinan 250100China

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bettman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of E1 Karoui, Kapoudjian, Pardoux, Peng and Quenez （1997）, which turns out to be very useful because it allows us to estimate the LP-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.

关键词： stochastic differential games value function reflected backward stochastic differential equations,dynamic programming principle Isaacs equations with obstacles viscosity solution

Inf-convolution of G-expectations

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Science China Mathematics 2010年第8期53卷 1957-1970页

作者： buckdahn rainer Laboratoire de Math'ematiques CNRS-UMR 6205Universite' de Bretagne Occidentale

In this paper we will discuss the optimal risk transfer problems when risk measures are generated by G-expectations,and we present the relationship between inf-convolution of G-expectations and the infconvolution of d... 详细信息

关键词： inf-convolution G-expectation G-normal distribution G-Brownian motion

Peng's maximum principle for a stochastic control problem driven by a fractional and a standard Brownian motion

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Science China Mathematics 2014年第10期57卷 2025-2042页

作者： buckdahn rainer JING Shuai Departement de Mathematiques Universite de Bretagne Occidentale School of Mathematics Shandong University School of Management Science and Engineering Central University of Finance and Economics

We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/*** apply an anticipative Girsanov transformation to transform the system into another one,driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian *** derive a maximum principle and the associated stochastic variational inequality,which both are generalizations of the classical case.

关键词： fractional Brownian motion stochastic control system backward stochastic differential equation variational inequality maximum principle Girsanov transformation Galtchouk-Kunita-Watanabe decomposition

Fully nonlinear stochastic and rough PDEs:Classical and viscosity solutions

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Probability, Uncertainty and Quantitative Risk 2020年第1期5卷 154-212页

作者： rainer.buckdahn Christian Keller Jin Ma Jianfeng Zhang Univ Brest UMR CNRS 6205Laboratoire de Mathematiques de Bretagne AtlantiqueBrest´Franceand Shandong UniversityJinanChina Department of Mathematics University of Central FloridaOrlandoFloridaUnited States Department of Mathematics University of Southern CaliforniaLos AngelesCaliforniaUnited States

We study fully nonlinear second-order(forward)stochastic *** can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified *** the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test *** notion of viscosity solutions is equivalent to the alternative using ***,we prove basic properties such as consistency,stability,and a partial comparison principle in the general *** the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.

关键词： Stochastic PDEs path-dependent PDEs rough PDEs rough paths viscosity solutions comparison principle functional Ito formulaˆ characteristics rough Taylor expansion

On the compensator of the default process in an information-based model

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Probability, Uncertainty and Quantitative Risk 2017年第1期2卷 230-250页

作者： Matteo Ludovico Bedini rainer.buckdahn Hans-Jurgen Engelbert Numerix MilanoItaly Universite de Bretagne Occidentale BrestFrance´School of MathematicsShandong UniversityJinanShandong ProvincePeople’s Republic of China Friedrich-Schiller-Universitat Fakult¨at f¨ur Mathematik und InformatikInstitut f¨ur StochastikJena¨Germany

This paper provides sufficient conditions for the time of bankruptcy(of a company or a state)for being a totally inaccessible stopping time and provides the explicit computation of its compensator in a framework where the flow of market information on the default is modelled explicitly with a Brownian bridge between 0 and 0 on a random time interval.

关键词： Default time Totally inaccessible stopping time Brownian bridge on random intervals Local time Credit risk Compensator process

Special issue dedicated to Alain Bensoussan on the occasion of his 80th birthday:Preface

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Probability, Uncertainty and Quantitative Risk 2022年第3期7卷 I0003-I0004页

作者： buckdahn rainer Juan Li Shige Peng 不详

We are delighted to present this special issue of PUQR in honor of Professor Alain Bensoussan on the occasion of his 80th *** this birthday provides a good opportunity to celebrate the life and the successes of an outstanding researcher,the COVID-19 epidemic has made it hard for a normal *** hope that this special issue of collected papers will nevertheless provide a lasting mark for his birthday and express the appreciation and best wishes to Alain Bensoussan,from his colleagues and former students,from his co-authors and co-co-authors around the world,for a long life in good health and creative power.

关键词： birthday occasion Ala

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Editorial

Probability, Uncertainty and Quantitative Risk 2016年第1期1卷 108-110页

作者： Shige Peng rainer.buckdahn Juan Li Shandong University Ji’nanChina Universitéde Bretagne Occidentale BrestFrance Shandong University WeihaiChina

Dear All,It is with great pleasure that we welcome you to the first issue of our journal,PUQR–Probability,Uncertainty and Quantitative Risk,a peer-reviewed openaccess *** its recent and very dynamic development,the theory of backward stochastic differential equations has attracted many researchers,with its vast field of applications in stochastic control,games,finance,and deterministic and stochastic partial differential *** has spurred the development of new areas for research such as nonlinear dynamic expectation theory,e.g.,g and G-expectation,and path-dependent partial differential equations,while also finding new applications for problems of ambiguity,uncertainty,quantitative risk,and recursive utility in finance and *** we further this field,it is important to provide a forum to stimulate future development with a journal that focuses on these *** precisely。

关键词： stochastic expectation backward

Friction between a temperature dependent viscoelastic body and a rough surface

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Friction 2016年第1期4卷 50-64页

作者： rainer HEISE Institute for Mechanics Technische Universit(a)t Berlin Straβe des 17.Juni 135 Berlin D-10623 Germany

In this study, we investigate the friction between a one-dimensional elastomer and a one-dimensional rigid randomly rough surface. Special emphasis is laid on the temperature dependence of the elastomer and its effect on the frictional behavior of the contact. The elastomer is modeled as a Kelvin body in a one-dimensional substitute model in the spirit of the method of dimensionality reduction. The randomly rough surface is a self-affine one-dimensional fractal. We provide a short discussion of a conical indenter pressed in a displacement controlled process into an elastomer. These analytical considerations are taken as a basis for the treatment of the randomly rough counter surface in contact to an elastomer with and without temperature dependent viscosity. We identify dimensionless quantities describing this process, introduce a thermal length scale, and give estimates for the coefficient of friction as function of velocity, indentation and thermal quantities.

关键词： elastomer friction temperature dependence coefficient of friction

Influence of slant of objective on image formation in optical microscopes

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Chinese Optics Letters 2008年第8期6卷 603-606页

作者： rainer Kning Physikalisch-Technische Bundesanstalt Braunschweig Bundesallee 10038116 BraunschweigGermany

A microscope image formation model based on scalar diffraction and Fourier optics has been developed, which takes a slant angle between the optical axis and the observed surface into account. The theoretical investigations of the imaging of line structures using this model show that reflection type microscopes are much stronger influenced by the slant angle than transmission type microscopes. In addition, the slant angle changes the image contrast and the image shape of a line structure, especially its edge. The larger the slant angle, the stronger the decrease of the image contrast, and the less steep the edge slope in both types of microscopes. Furthermore, the larger the numerical aperture of the objective, the less the effect of the slant angle on the line image shape.

关键词： line this