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| 1 | Survey on normal distributions,central limit theorem,Brownian motion and the related stochastic calculus under sublinear expectations显示文摘This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It's type.The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk,statistics and other industrial problems. | PENG ShiGe Institute of Mathematics,Shandong University,Jinan 250100,China | 2009 | Science China Mathematics2009，52，7： | 54 |
| 2 | NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS显示文摘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. | PENG Shige | 2005 | Chinese Annals of Mathematics,Series B2005，26，2： | 27 |
| 3 | BSDE,path-dependent PDE and nonlinear Feynman-Kac formula显示文摘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 ] × Rd. 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 FeynmanKac formula for a general non-Markovian BSDE. Some main properties of solutions of this new PDEs are also obtained. | PENG ShiGe WANG FaLei | 2016 | Science China Mathematics2016，59，1： | 9 |
| 4 | Stochastic calculus with respect to G-Brownian motion viewed through rough paths显示文摘We study rough path properties of stochastic integrals of Ito's type and Stratonovich's type with respect to G-Brownian motion. The roughness of G-Brownian motion is estimated and then the pathwise Norris lemma in G-framework is obtained. | PENG ShiGe ZHANG HuiLin | 2017 | Science China Mathematics2017，60，1： | 2 |
| 5 | Reflected solutions of backward stochastic differential equations driven by G-Brownian motion显示文摘In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a 'martingale condition' instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty. | Hanwu Li Shige Peng Abdoulaye Soumana Hima | 2018 | Science China Mathematics2018，61，1： | 2 |
| 6 | SMALLEST g-SUPERSOLUTION FOR BSDE WITH CONTINUOUS DRIFT COEFFICIENTS显示文摘The authors prove the existence and uniqueness of smallest g-supersolution with an equality constrains on (y, z) for one demensional stochastic differential equations whose drift coefficients are continuous and linearly growing, and whose terminal conditions are square integrable. | LIN QINGQUAN (School of Mathematics and System Science, Shandong University, Jinan 250100, China.) PENG SHIGE(School of Mathematics and System Science, Shandong University, Jinan 250100, China.) E-mail: Linqingquan@math. sdu. cn Peng@sdu. edu. cn | 2000 | Chinese Annals of Mathematics,Series B2000，21，3： | 2 |
| 7 | Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation显示文摘 | PENG Shige | 2008 | Stochastic Processes and Applications2008，118，12： | 1 |
| 8 | Adapted solution of a backward stochastic differential equations显示文摘 | Pardoux E Peng Shige | 1990 | Systems Control Lett1990，14，1： | 1 |
| 9 | Multi - dimensional < i > G - Brownian motion and related stochastic calculus under < i > G - expectation 显示文摘 | Peng Shige | 2008 | Stochastic Processes and Their Applications2008，118，12： | 1 |
| 10 | Continuous properties of g-martingales显示文摘 | Chen Zengjing Peng Shige | 2001 | Chin Ann of Math2001，22，1： | 1 |
| 11 | Backward doubly stochastic differential equations and systems of quasilinear SPDEs显示文摘 | Etienne Pardoux Shige Peng | 1994 | Probability Theory and Related Fields1994，，2： | 1 |
| 12 | Backward stochastic differential equations and application to optimal control显示文摘 | PENG Shige | 1993 | Appl Math Optim1993，27，： | 1 |
| 13 | Adapted solution of a backward stochastic equation显示文摘 | PARDOUX E PENG Shige | 1990 | Systems & Control Letters1990，14，1： | 1 |
| 14 | Backward Stochastic Differential Equations and Ap- plications to Optimal Control 显示文摘 | Shige Peng | 1993 | Aplied Mathematics & Op- timization1993，，27： | 1 |
| 15 | Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyers type显示文摘 | Shige Peng | 1999 | Probability Theory and Related Fields1999，，4： | 1 |
| 16 | A stability theorem of backward stochastic differential equations and its application显示文摘 | Ying Hu Shige Peng | 1997 | Comptes Rendus de l’Academie des Sciences Series I Mathematics1997，，9： | 1 |
| 17 | Adapted solution of a backward stochastic differential equation显示文摘 | Pardoux E Peng Shige | | 0，，01： | 1 |
| 18 | A decomposition theorem of g-martingales显示文摘 | Chen Zengjing Peng Shige | | 0，，02： | 1 |
| 19 | Adapted solution of a backward stochastic differential equation显示文摘 | Pardoux E Peng Shige | 1990 | Systems Control Lett1990，14，1： | 1 |
| 20 | Adapted solution of a backward stochastic differential equation显示文摘 | Pardoux E Peng Shige | 1990 | Systems Control Lett1990，14，1： | 1 |
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