摘要翻译：
本文研究了由条件$G$-期望引起的多维动态风险测度。提出了多维$G$-期望的概念，以提供非线性期望的多维版本。利用多维倒向随机微分方程的比较定理、唯一性定理和矩形解的存在性的显式的技术结果，给出了多维条件期望和多维动态风险测度的恒常性、单调性、正性、齐性和可译性的充要条件；证明了多维动态$G$-风险测度是非渐凸的当且仅当生成元$G$满足拟单调渐凸条件。给出了多维动态凸$G$-风险测度的一般对偶表示，其中惩罚项的表达更为精确。结果表明，模型不确定性导致风险测度的凸性。在应用方面，我们展示了如何将这一多维方法应用于度量有相互影响的子公司的公司的破产风险；研究了$\protect\gamma$-容忍$G$-风险度量的最优风险分担。本文最后还研究了保险$G$-风险测度和其他诱导$G$-风险测度的方法。
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英文标题：
《Multidimensional dynamic risk measure via conditional g-expectation》
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作者：
Yuhong Xu
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最新提交年份：
2012
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Risk Management        风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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英文摘要：
  This paper deals with multidimensional dynamic risk measures induced by conditional $g$-expectations. A notion of multidimensional $g$-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, homogeneity and translatability properties of multidimensional conditional $g$-expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic $g$-risk measure is nonincreasingly convex if and only if the generator $g$ satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex $g$-risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacted subsidiaries; optimal risk sharing for $\protect\gamma $-tolerant $g$-risk measures is investigated. Insurance $g$-risk measure and other ways to induce $g$-risk measures are also studied at the end of the paper.
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PDF链接：
https://arxiv.org/pdf/1011.3685