This article studies quadratic semimartingale BSDEs arising in power utility maximization when the market price of risk is of BMO type. In a Brownian setting we provide a necessary and sufficient condition for the existence of a solution but show that uniqueness fails to hold in the sense that there exists a continuum of distinct square-integrable solutions. This feature occurs since, contrary to the classical Ito representation theorem, a representation of random variables in terms of stochastic exponentials is not unique. We study in detail when the BSDE has a bounded solution and derive a new dynamic exponential moments condition which is shown to be the minimal sufficient condition in a general filtration. The main results are complemented by several interesting examples which illustrate their sharpness as well as important properties of the utility maximization BSDE.
The reverse Hölder inequality for matrix-valued stochastic exponentials and applications to quadratic BSDE systems
本文研究了矩阵值鞅的逆赫尔德不等式、系数无界的线性倒向随机微分方程的适定性和二次BSDES系统的适定性这三个概念之间的联系。特别地，我们证明了具有bmo（bounded mean oscillation）系数的线性鞅是适定性的当且仅当相关矩阵值鞅的随机指数满足逆赫尔德不等式.此外，我们给出了满足这两个等价条件的结构条件。最后，我们将所得结果应用到线性方程上，得到了两类新的具有特殊结构的非马尔可夫二次Bundle系统的全局适定性结果。
Stability of exponential utility maximization with respect to market perturbations
研究了市场夏普比率扰动下期望指数效用最大化的连续性。通过只关注连续性，我们施加较弱的正则性条件比文献中发现的。具体地说，除了Larsen and Schmidtković（2007）（ArXiv：0706.0474）的$V$ -紧性假设之外，我们还需要一个局部$bmo$假设，这个条件在Larsen and Schmidtković（2007）的设定中总是被平凡地满足。对于形式为$S = M + \int λd<M>$的市场，这些条件同时隐含在合适的$bmo$空间中$λ\cdot M$的范数上存在一致的界。
Quadratic Mean-Field BSDEs and Exponential Utility Maximization
School of Mathematics and Statistics, The University of Sydney（悉尼大学数学与统计学院）；School of Mathematics, Monash University（蒙纳士大学数学学院）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）；Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学非线性期望前沿科学中心（教育部））；Shandong Province Key Laboratory of Financial Risk, Shandong University（山东大学山东省金融风险重点实验室）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰证券金融研究院）
Multidimensional quadratic BSDEs with weak interactions and their applications in mean-field games of controls
Humboldt University Berlin（柏林洪堡大学）；Research Center for Mathematics and Interdisciplinary Sciences, Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学数学与交叉科学研究中心，非线性期望前沿科学中心（教育部））
