The objective of this paper is to study the local time and Tanaka formula of symmetric G-martingales.We introduce the local time of G-martingales and show that it belongs to the G-expectation space LG^2(ΩT).By a loca...The objective of this paper is to study the local time and Tanaka formula of symmetric G-martingales.We introduce the local time of G-martingales and show that it belongs to the G-expectation space LG^2(ΩT).By a localization argument,we obtain the bicontinuous modification of local time.Furthermore,we give the Tanaka formula for convex functions of G-martingales.展开更多
In this article, a sublinear expectation induced by G-expectation is introduced, which is called G- evaluation for convenience. As an application, we prove that for any ξ∈ L β G (Ω T ) with some β > 1 the mart...In this article, a sublinear expectation induced by G-expectation is introduced, which is called G- evaluation for convenience. As an application, we prove that for any ξ∈ L β G (Ω T ) with some β > 1 the martingale decomposition theorem under G-expectaion holds, and that any β > 1 integrable symmetric G-martingale can be represented as an Ito integral w.r.t. G-Brownian motion. As a byproduct, we prove a regularity property for G-martingales: Any G-martingale {M t } has a quasi-continuous version.展开更多
The authors study the continuous properties of square integrable g-martingales via backward stochastic differential equations (shortly BSDEs) and get a general upcrossing inequality and an optional stopping theorem fo...The authors study the continuous properties of square integrable g-martingales via backward stochastic differential equations (shortly BSDEs) and get a general upcrossing inequality and an optional stopping theorem for g-martingales.展开更多
This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectatio...This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectations via nonlinear Markov chains. Com- pared to the author’s previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probabil- ity measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations. The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures.展开更多
In this paper,the authors consider a reflected backward stochastic differential equation driven by a G-Brownian motion(G-BSDE for short),with the generator growing quadratically in the second unknown.The authors obtai...In this paper,the authors consider a reflected backward stochastic differential equation driven by a G-Brownian motion(G-BSDE for short),with the generator growing quadratically in the second unknown.The authors obtain the existence by the penalty method,and some a priori estimates which imply the uniqueness,for solutions of the G-BSDE.Moreover,focusing their discussion at the Markovian setting,the authors give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.展开更多
基金Supported by the National Natural Science Foundation of China(No.11601282)the Natural Science Foundation of Shandong Province(No.ZR2016AQ10)
文摘The objective of this paper is to study the local time and Tanaka formula of symmetric G-martingales.We introduce the local time of G-martingales and show that it belongs to the G-expectation space LG^2(ΩT).By a localization argument,we obtain the bicontinuous modification of local time.Furthermore,we give the Tanaka formula for convex functions of G-martingales.
基金supported by National Basic Research Program of China (973 Program) (Grant No. 2007CB814902)
文摘In this article, a sublinear expectation induced by G-expectation is introduced, which is called G- evaluation for convenience. As an application, we prove that for any ξ∈ L β G (Ω T ) with some β > 1 the martingale decomposition theorem under G-expectaion holds, and that any β > 1 integrable symmetric G-martingale can be represented as an Ito integral w.r.t. G-Brownian motion. As a byproduct, we prove a regularity property for G-martingales: Any G-martingale {M t } has a quasi-continuous version.
基金Project supported by the National Natural Science Foundation of China (No.79790130).
文摘The authors study the continuous properties of square integrable g-martingales via backward stochastic differential equations (shortly BSDEs) and get a general upcrossing inequality and an optional stopping theorem for g-martingales.
基金Project supported by the National Natural Science Foundation of China(No.10131040).
文摘This paper deals with nonlinear expectations. The author obtains a nonlinear gen- eralization of the well-known Kolmogorov’s consistent theorem and then use it to con- struct ?ltration-consistent nonlinear expectations via nonlinear Markov chains. Com- pared to the author’s previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probabil- ity measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations. The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures.
基金supported by the National Science Foundation of China(No.11631004)the Science and Technology Commission of Shanghai Municipality(No.14XD1400400).
文摘In this paper,the authors consider a reflected backward stochastic differential equation driven by a G-Brownian motion(G-BSDE for short),with the generator growing quadratically in the second unknown.The authors obtain the existence by the penalty method,and some a priori estimates which imply the uniqueness,for solutions of the G-BSDE.Moreover,focusing their discussion at the Markovian setting,the authors give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.
