非线性期望及其在金融中的应用

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, we consider the existence and uniqueness question of BSDEs driven by fractional brownian motions. Since the theory of GBSDEs have many important applications in mathematical finance area, then we wants to get some fundational results in this fields.

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。

在G布朗运动和G期望框架下，申请人研究了GBSDE生成元的表示定理以及在GBSDE所定义的非线性期望下研究相应的G凸函数相关的性质和等价性结果。前一个结果已经发表SCI论文，后面一个结果还在审稿过程中。.在G布朗运动框架下，利用粗轨道理论研究了对应定义的局部时问题，并将此结果应用与推广对应的Ito公式。该结果还在审稿过程中。.在混合分数布朗运动方面申请人和合作者研究了针对满足一类可积性且其空间构成Banach空间的函数f，讨论了其对应定义的平方变差过程的存在性，并证明对于F'=f并且绝对连续的F，有Ito公式成立。该结果发表SCI论文。.申请人研究了一类由Levy过程驱动的随机微分方程解的问题，用来刻画一种不完备金融市场的期权定价问题。这类由Levy过程驱动的随机微分方程其利率和波动率都是依赖于资产价格的函数，利用Follmer-Schweizer最小鞅测度方法找出对应的最小鞅测度，得到该模型对应的欧式期权定价公式。该结果发表核心期刊论文。.项目第一参与者主要研究了在G期望框架下随机变量的拟连续性问题，以及动态规划原理，以及G扩散过程的遍历性问题，相应发表了3篇SCI论文。