It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In this work we show that in a Brownian filtration the “Optimized Certainty Equivalent” risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This can be seen as a substitute of sorts whenever they lack time consistency, and covers the cases of conditional value-at-risk and monotone mean-variance. Our method consists of focusing on the convex dual representation, which suggests an expression of the risk measure as the value of a stochastic control problem on an extended the state space. With this we can obtain a dynamic programming principle and use stochastic control techniques, along with the theory of viscosity solutions, which we must adapt to cover the present singular situation.

中文翻译：

---

关于布朗滤波中一些时间不一致风险度量的动态表示

从Kupper和Schachermayer的工作中众所周知，大多数定律的风险度量都不是时间一致的，因此不接受将动态表示作为反向随机微分方程。在这项工作中，我们表明，在布朗过滤中，可以通过PDE技术（即动态地）计算Ben-Tal和Teboulle的“最佳确定性当量”风险度量。每当它们缺乏时间一致性时，就可以将其视为替代方法，并涵盖条件风险值和单调均值方差的情况。我们的方法着重于凸对偶表示，这建议将风险度量表示为在扩展状态空间上的随机控制问题的值。这样，我们可以获得动态的编程原理并使用随机控制技术，