英文标题：
《Risk measures and progressive enlargement of filtration: a BSDE approach》
---
作者：
Alessandro Calvia and Emanuela Rosazza Gianin
---
最新提交年份：
2020
---
英文摘要：
  We consider dynamic risk measures induced by Backward Stochastic Differential Equations (BSDEs) in enlargement of filtration setting. On a fixed probability space, we are given a standard Brownian motion and a pair of random variables $(\\tau, \\zeta) \\in (0,+\\infty) \\times E$, with $E \\subset \\mathbb{R}^m$, that enlarge the reference filtration, i.e., the one generated by the Brownian motion. These random variables can be interpreted financially as a default time and an associated mark. After introducing a BSDE driven by the Brownian motion and the random measure associated to $(\\tau, \\zeta)$, we define the dynamic risk measure $(\\rho_t)_{t \\in [0,T]}$, for a fixed time $T > 0$, induced by its solution. We prove that $(\\rho_t)_{t \\in [0,T]}$ can be decomposed in a pair of risk measures, acting before and after $\\tau$ and we characterize its properties giving suitable assumptions on the driver of the BSDE. Furthermore, we prove an inequality satisfied by the penalty term associated to the robust representation of $(\\rho_t)_{t \\in [0,T]}$ and we discuss the dynamic entropic risk measure case, providing examples where it is possible to write explicitly its decomposition and simulate it numerically.
---
中文摘要：
我们考虑了在扩大过滤设置时由倒向随机微分方程（BSDE）引起的动态风险度量。在固定概率空间上，我们给出了一个标准的布朗运动和一对随机变量$（\\tau，\\zeta）\\in（0，+\\infty）\\乘以E$，以及$E\\subset\\mathbb{R}^m$，它们扩大了参考过滤，即由布朗运动生成的过滤。这些随机变量在财务上可以解释为默认时间和相关标记。在引入由布朗运动驱动的BSDE和与$（\\tau，\\zeta）$相关的随机测度后，我们定义了由其解诱导的固定时间$t>0$的动态风险测度$（\\rho\\u t）{t\\in[0，t]}$。我们证明了$（\\rho\\u t）{t\\in[0，t]}$可以分解为一对风险度量，在$\\tau$之前和之后，我们刻画了它的性质，并对BSDE的驱动因素给出了适当的假设。此外，我们证明了与$（\\rho\\u t）{t\\in[0，t]}$的鲁棒表示相关的惩罚项满足的不等式，并讨论了动态熵风险度量情况，提供了可以显式编写其分解并进行数值模拟的示例。
---
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
--
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
--
一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
--

---
PDF下载：
-->