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Optimal Control of Conditional Value-at-Risk in Continuous Time

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  • Christopher W. Miller
  • Insoon Yang

Abstract

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.

Suggested Citation

  • Christopher W. Miller & Insoon Yang, 2015. "Optimal Control of Conditional Value-at-Risk in Continuous Time," Papers 1512.05015, arXiv.org, revised Jan 2017.
  • Handle: RePEc:arx:papers:1512.05015
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    References listed on IDEAS

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    Cited by:

    1. Avinash N. Madavan & Subhonmesh Bose, 2021. "A Stochastic Primal-Dual Method for Optimization with Conditional Value at Risk Constraints," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 428-460, August.
    2. van Staden, Pieter M. & Dang, Duy-Minh & Forsyth, Peter A., 2021. "The surprising robustness of dynamic Mean-Variance portfolio optimization to model misspecification errors," European Journal of Operational Research, Elsevier, vol. 289(2), pages 774-792.
    3. Masashi Ieda, 2021. "Continuous-time Portfolio Optimization for Absolute Return Funds," Papers 2108.09985, arXiv.org, revised Mar 2022.
    4. Seungki Min & Ciamac C. Moallemi & Costis Maglaras, 2022. "Risk-Sensitive Optimal Execution via a Conditional Value-at-Risk Objective," Papers 2201.11962, arXiv.org.

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