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Linear-Quadratic Stochastic Delayed Control and Deep Learning Resolution

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  • William Lefebvre

    (BNP Paribas Global Markets
    Université de Paris and Sorbonne Université)

  • Enzo Miller

    (Université de Paris and Sorbonne Université)

Abstract

We consider a simple class of stochastic control problems with a delayed control, in both the drift and the diffusion part of the state stochastic differential equation. We provide a new characterization of the solution in terms of a set of Riccati partial differential equations. Existence and uniqueness of a solution are obtained under a sufficient condition expressed directly as a relation between the time horizon, the drift, the volatility and the delay. Furthermore, a deep learning scheme (The code is available in a IPython notebook .) is designed and used to illustrate the effect of the delay feature on the Markowitz portfolio allocation problem with execution delay.

Suggested Citation

  • William Lefebvre & Enzo Miller, 2021. "Linear-Quadratic Stochastic Delayed Control and Deep Learning Resolution," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 134-168, October.
    • Handle: RePEc:spr:joptap:v:191:y:2021:i:1:d:10.1007_s10957-021-01923-x
    DOI: 10.1007/s10957-021-01923-x
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    References listed on IDEAS

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