Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general This paper proves that Jensen's inequality for g-expectation holds in general if...Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general This paper proves that Jensen's inequality for g-expectation holds in general if and only if the generator g (t, z) is super-homogeneous in z. In particular, g is not necessarily convex in z.展开更多
A kind of discounted problems for singular diffusion control have been studied. The drift and diffusion coefficients of state process are nonlinear. The class of models has been basically extended from a corresponding...A kind of discounted problems for singular diffusion control have been studied. The drift and diffusion coefficients of state process are nonlinear. The class of models has been basically extended from a corresponding one established by Karatzes et al. before. By applying some analysis methods different from earlier works, the sufficient and necessary conditions of the existence of optimal control have been obtained. If an optimal control exists, it is a 'transient reflection' of state process for two lines.展开更多
In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus $ \mathbb{T}^2 $ perturbed by a Lévy process. The existence of invariant me...In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus $ \mathbb{T}^2 $ perturbed by a Lévy process. The existence of invariant measure of the solutions are proved also.展开更多
In this paper the boundedness of a derived function of a solution about a class of diffusion variational equations is discussed. The application of it to related stochastic analysis problems is also illustrated. What ...In this paper the boundedness of a derived function of a solution about a class of diffusion variational equations is discussed. The application of it to related stochastic analysis problems is also illustrated. What should be emphasized is that the problem discussed and the ways proved in this paper are fundamentally new and the conclusion of this paper is fairly profound.展开更多
Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen＇s inequality holds for backward stochastic differential equations with generator g if and only if g is independent o...Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen＇s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g（t, 0）≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen＇s inequality for gexpectation in [4, 7-9].展开更多
In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimen...In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations （BSDEs） driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic （LQ） optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.展开更多
We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. Thi...We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman- Kac formula for a general non-Markoviau BSDE. Some main properties of solutions of this new PDEs are also obtained.展开更多
Motivated by a duopoly game problem,the authors study an optimal control problem where the system is driven by Brownian motion and Poisson point process and has elephant memory for the control variable and the state v...Motivated by a duopoly game problem,the authors study an optimal control problem where the system is driven by Brownian motion and Poisson point process and has elephant memory for the control variable and the state variable.Firstly,the authors establish the unique solvability of an anticipated backward stochastic differential equation,derive a stochastic maximum principle,and prove a verification theorem for the aforementioned optimal control problem.Furthermore,the authors generalize these results to nonzero-sum stochastic differential game problems.Finally,the authors apply the theoretical results to the duopoly game problem and obtain the corresponding Nash equilibrium solution.展开更多
Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation c...Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation characterizes the path-independence property of the density process of Girsanov transformation for the stochastic differential equation.Our assertion also holds for SDEs on a connected differential manifold.展开更多
This paper applies stochastic evolutionary game theory to analyzing the stability of cooperation among members against external opportunism in a multi-firm alliance.The authors first review the pros and cons of pertin...This paper applies stochastic evolutionary game theory to analyzing the stability of cooperation among members against external opportunism in a multi-firm alliance.The authors first review the pros and cons of pertinent traditional models,and then a stochastic game model on decisions is proposed,where a coordination parameter,a time variable,a punishment effect and bounded rationality are considered.The Gauss white noise is introduced to reflect the random disturbance in the process.Several sufficient criteria on stability are developed,which enable us to investigate"if-then"type scenarios and project the impact of different strategies.展开更多
基金Project supported by the National Natural Science Foundation of China (No.10131030)
文摘Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general This paper proves that Jensen's inequality for g-expectation holds in general if and only if the generator g (t, z) is super-homogeneous in z. In particular, g is not necessarily convex in z.
文摘A kind of discounted problems for singular diffusion control have been studied. The drift and diffusion coefficients of state process are nonlinear. The class of models has been basically extended from a corresponding one established by Karatzes et al. before. By applying some analysis methods different from earlier works, the sufficient and necessary conditions of the existence of optimal control have been obtained. If an optimal control exists, it is a 'transient reflection' of state process for two lines.
基金supported by National Basic Research Program of China (Grant No. 2006CB8059000)Science Fund for Creative Research Groups (Grant No.10721101)+2 种基金National Natural Science Foundation of China (GrantNos.10671197,10671168)Science Foundation of Jiangsu Province (Grant Nos.BK2006032,06-A-038,07-333)Key Lab of Random Complex Structures and Data Science,Chinese Academy of Sciences
文摘In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus $ \mathbb{T}^2 $ perturbed by a Lévy process. The existence of invariant measure of the solutions are proved also.
基金a grant from National Natural Science Foundation of China (10671072)Doctoral Program Foundation of the Ministry of Education of China (20060269016)+1 种基金the National Basic Research Program (973 Program,2007CB814904) of Chinathe NSF of Anhui Educational Bureau (KJ2008B243)
基金Project supported by National Science Foundation
文摘In this paper the boundedness of a derived function of a solution about a class of diffusion variational equations is discussed. The application of it to related stochastic analysis problems is also illustrated. What should be emphasized is that the problem discussed and the ways proved in this paper are fundamentally new and the conclusion of this paper is fairly profound.
基金Project supported by the National Natural Science Foundation of China (No.10325101)the Science Foundation of China University of Mining and Technology.
文摘Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen＇s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g（t, 0）≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen＇s inequality for gexpectation in [4, 7-9].
基金This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2007CB814904the Natural Science Foundation of China under Grant No. 10671112+1 种基金Shandong Province under Grant No. Z2006A01Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20060422018
文摘In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations （BSDEs） driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic （LQ） optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.
基金supported by National Natural Science Foundation of China(Grant No.10921101)the Programme of Introducing Talents of Discipline to Universities of China(Grant No.B12023)the Fundamental Research Funds of Shandong University
文摘We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman- Kac formula for a general non-Markoviau BSDE. Some main properties of solutions of this new PDEs are also obtained.
基金supported by the National Key R&D Program of China under Grant No.2022YFA1006103the National Natural Science Foundation of China under Grant Nos.61821004,61925306,11831010,71973084,61977043the National Science Foundation of Shandong Province under Grant Nos.ZR2019ZD42 and ZR2020ZD24。
文摘Motivated by a duopoly game problem,the authors study an optimal control problem where the system is driven by Brownian motion and Poisson point process and has elephant memory for the control variable and the state variable.Firstly,the authors establish the unique solvability of an anticipated backward stochastic differential equation,derive a stochastic maximum principle,and prove a verification theorem for the aforementioned optimal control problem.Furthermore,the authors generalize these results to nonzero-sum stochastic differential game problems.Finally,the authors apply the theoretical results to the duopoly game problem and obtain the corresponding Nash equilibrium solution.
基金supported by Laboratory of Mathematics and Complex Systems,National Natural Science Foundation of China(Grant No.11131003)Specialized Research Fund for the Doctoral Program of Higher Educationthe Fundamental Research Funds for the Central Universities
文摘Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation characterizes the path-independence property of the density process of Girsanov transformation for the stochastic differential equation.Our assertion also holds for SDEs on a connected differential manifold.
基金supported by the Scientific Research Foundation of Shandong Province in 2014the Outstanding Young Scientist Award under Grant No.BS2014SF009+1 种基金the National Natural Science Research of China under Grant Nos.71373194,71101059,71172086,71272122,61304175the Ministry of Education of Humanities and Social Science Youth Fund Project under Grant No.13YJC630013
文摘This paper applies stochastic evolutionary game theory to analyzing the stability of cooperation among members against external opportunism in a multi-firm alliance.The authors first review the pros and cons of pertinent traditional models,and then a stochastic game model on decisions is proposed,where a coordination parameter,a time variable,a punishment effect and bounded rationality are considered.The Gauss white noise is introduced to reflect the random disturbance in the process.Several sufficient criteria on stability are developed,which enable us to investigate"if-then"type scenarios and project the impact of different strategies.
