非线性期望及其在金融中的应用

项目名称： 非线性期望及其在金融中的应用

项目编号： No.11301068

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 何坤

作者单位： 东华大学

项目金额： 22万元

中文摘要： g-期望是彭实戈院士1997首次在Wiener测度的框架下引入的一种动态相容的非线性期望。彭院士又于2004年在对股票波动率的不确定性问题研究的基础上提出了G-期望和G-布朗运动，G-期望可以用于处理奇异概率族下的金融问题。2012年他同胡明尚，嵇少林，宋永生进一步建立了由G-布朗运动驱动的倒向随机微分方程（GBSDEs）理论。本项目旨在研究如下几个问题：从GBSDE解到生成元方向的逆比较定理；GBSDE解满足次可加性、正其次性、凸性和保常数性时同生成元之间的等价性条件；GBSDE解的Jensen不等式和G-凸函数的研究；GBSDE解同Choquet-期望之间的关系；以及分数布朗运动驱动的GBSDE解的存在唯一性。GBSDE理论在金融中有重要应用，我们希望将得到的理论结果用于奇异概率族下的金融问题中。

中文关键词： G布朗运动；G期望；GBSDE；混合分数布朗运动；

英文摘要： g-Expectation is a kind of dynamic nonlinear expectation that was introduced by professor Peng in 1997 under the Wiener measure. Considering the volatility uncertainty of financial market, professor Peng introduced G-expectation and G-Brownian motion in 2004. The G-expectation deals with a group of singular probability measures in financial questions. Recently, the existence and uniqueness of Backward Stochastic Differential Equations dirven by G-Brownian motions (GBSDEs) was solved and a comparison result have been proved after that. This project is to study several questions in this fields: First we consider a converse comparison problem of GBSDEs. Second We study equivalent conditions between the solution of GBSDE and generator g, when the g-expectation satisfies sublinearity, positive homogenity, convexity and translation invariant property, respectively. Third we considering the Jensen inequality of GBSDE and the G-convex function question. Forth we define one Choquet capacity by a given g-expectation, that was an important concept in mathematical economics. And we derive a nonlinear expectation--Choquet-expectation from the Choquet capacity. Then we talking about the relationship between the group of Choquet-expectations and the group of g-expectations, if there have any intersections or not. Finally, w

英文关键词： G Brownian Motion；G expectation；GBSDE；Mixed fractional Brownian Motion；