Optimal Control of Conditional Value-at-Risk in Continuous Time
Optimal Control of Conditional Value-at-Risk in Continuous Time
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DOI：
10.1137/16m1058492
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发表时间：
2015-12
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通讯作者：
Christopher W. Miller;Insoon Yang
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文献类型：
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作者：
Christopher W. Miller;Insoon Yang
考虑目标中具有条件风险值（CVaR）的连续时间随机最优控制问题。这些问题的主要困难来自于时间不一致，这使我们无法直接使用动态规划。为了解决这一挑战，我们将其转化为一个等效的双层优化问题，其中内部优化问题是标准随机控制。进一步给出了外目标函数凸可微的条件。我们通过Hamilton-Jacobi-Bellman方程计算外部目标的值，并通过线性抛物方程的粘度解计算其梯度，从而允许我们执行梯度下降。该结果的意义在于，我们在不提升状态空间的情况下，提供了一种有效的基于动态规划的CVaR最优控制算法。为了扩大所提出算法的适用性，我们在我们的关键假设不成立的情况下提出了收敛的近似方案，并描述了相关的次优性边界。此外，我们将我们的方法扩展到更一般的风险度量类，其中包括平均方差和中位数偏差。我们还演示了CVaR约束下投资组合优化的具体应用。我们的结果为解决基于时间不一致的cvar序列优化提供了一个有效的框架。
We consider continuous-time stochastic optimal control problems featuring Conditional Value-at-Risk (CVaR) in the objective. The major difficulty in these problems arises from time-inconsistency, which prevents us from directly using dynamic programming. To resolve this challenge, we convert to an equivalent bilevel optimization problem in which the inner optimization problem is standard stochastic control. Furthermore, we provide conditions under which the outer objective function is convex and differentiable. We compute the outer objective's value via a Hamilton-Jacobi-Bellman equation and its gradient via the viscosity solution of a linear parabolic equation, which allows us to perform gradient descent. The significance of this result is that we provide an efficient dynamic programming-based algorithm for optimal control of CVaR without lifting the state-space. To broaden the applicability of the proposed algorithm, we propose convergent approximation schemes in cases where our key assumptions do not hold and characterize relevant suboptimality bounds. In addition, we extend our method to a more general class of risk metrics, which includes mean-variance and median-deviation. We also demonstrate a concrete application to portfolio optimization under CVaR constraints. Our results contribute an efficient framework for solving time-inconsistent CVaR-based sequential optimization.