Stochastic Control of Optimized Certainty Equivalents
Stochastic Control of Optimized Certainty Equivalents
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DOI：
10.1137/21m1407732
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发表时间：
2020-01
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影响因子：
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通讯作者：
Julio D. Backhoff Veraguas;Max Reppen;Ludovic Tangpi
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文献类型：
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作者：
Julio D. Backhoff Veraguas;Max Reppen;Ludovic Tangpi
优化确定性当量（OCE）是从业者和学者广泛使用的一系列风险衡量标准。这主要是由于它的易处理性以及它包含重要示例的事实，包括熵风险度量和平均风险值。在这项工作中，我们考虑随机最优控制问题，其中客观标准由 OCE 风险度量给出，或者换句话说，受控扩散的风险最小化问题。由于 OCE 的时间常常不一致，因此出现了一个主要困难。尽管如此，通过扩大状态空间，我们在相当普遍的情况下实现了时间一致性的某种替代。这使我们能够推导出动态规划原理，从而恢复（风险中性）随机控制理论的核心结果。特别是，我们证明了风险最小化问题的价值可以通过 Hamilton-Jacobi-Bellman-Issacs 方程的粘度解来表征。我们在合适的技术条件下进一步确立了后者的独特性。
Optimized certainty equivalents (OCEs) is a family of risk measures widely used by both practitioners and academics. This is mostly due to its tractability and the fact that it encompasses important examples, including entropic risk measures and average value at risk. In this work we consider stochastic optimal control problems where the objective criterion is given by an OCE risk measure, or put in other words, a risk minimization problem for controlled diffusions. A major difficulty arises since OCEs are often time inconsistent. Nevertheless, via an enlargement of state space we achieve a substitute of sorts for time consistency in fair generality. This allows us to derive a dynamic programming principle and thus recover central results of (risk-neutral) stochastic control theory. In particular, we show that the value of our risk minimization problem can be characterized via the viscosity solution of a Hamilton--Jacobi--Bellman--Issacs equation. We further establish the uniqueness of the latter under suitable technical conditions.