SIAM Journal on Financial Mathematics, Volume 11, Issue 3, Page 815-848, January 2020.
We consider dynamic risk measures induced by backward stochastic differential equations (BSDEs) in an 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.

中文翻译：

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风险衡量和过滤的逐步扩大：BSDE方法

SIAM金融数学杂志，第11卷，第3期，第815-848页，2020年1月。
我们考虑在过滤设置的扩大中由后向随机微分方程（BSDE）引起的动态风险度量。在固定的概率空间上，我们得到标准的布朗运动和一对随机变量$（\ tau，\ zeta）\ in（0，+ \ infty）\ times E $，其中$ E \ subset \ mathbb {R } ^ m $，可以扩大参考滤波，即由布朗运动产生的参考滤波。这些随机变量可以从财务上解释为默认时间和相关标记。在引入由布朗运动和与$（\ tau，\ zeta）$相关的随机度量驱动的BSDE之后，我们定义了动态风险度量$（\ rho_t）_ {t \ in [0，T]} $，用于由其解引起的固定时间$ T> 0 $。我们证明$（\ rho_t）_ {t \ in [0，T]} $可以分解为一对风险度量，在$ \ tau $之前和之后执行，我们通过对BSDE驱动程序的适当假设来表征其特性。此外，我们证明了与$（\ rho_t）_ {t [in [0，T]} $的鲁棒表示相关的惩罚项所满足的不等式，并讨论了动态熵风险测度的情况，并提供了可能的示例明确地写出它的分解并进行数值模拟。