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Impulse control problem on finite horizon with execution delay

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  • Bruder, Benjamin
  • Pham, Huyên

Abstract

We consider impulse control problems in finite horizon for diffusions with decision lag and execution delay. The new feature is that our general framework deals with the important case when several consecutive orders may be decided before the effective execution of the first one. This is motivated by financial applications in the trading of illiquid assets such as hedge funds. We show that the value functions for such control problems satisfy a suitable version of dynamic programming principle in finite dimension, which takes into account the past dependence of state process through the pending orders. The corresponding Bellman partial differential equations (PDE) system is derived, and exhibit some peculiarities on the coupled equations, domains and boundary conditions. We prove a unique characterization of the value functions to this nonstandard PDE system by means of viscosity solutions. We then provide an algorithm to find the value functions and the optimal control. This easily implementable algorithm involves backward and forward iterations on the domains and the value functions, which appear in turn as original arguments in the proofs for the boundary conditions and uniqueness results.

Suggested Citation

  • Bruder, Benjamin & Pham, Huyên, 2009. "Impulse control problem on finite horizon with execution delay," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1436-1469, May.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:5:p:1436-1469
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    References listed on IDEAS

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    1. Vathana Ly Vath & Mohamed Mnif & Huyên Pham, 2007. "A model of optimal portfolio selection under liquidity risk and price impact," Finance and Stochastics, Springer, vol. 11(1), pages 51-90, January.
    2. Alvarez, Luis H. R. & Keppo, Jussi, 2002. "The impact of delivery lags on irreversible investment under uncertainty," European Journal of Operational Research, Elsevier, vol. 136(1), pages 173-180, January.
    3. Bayraktar, Erhan & Egami, Masahiko, 2007. "The effects of implementation delay on decision-making under uncertainty," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 333-358, March.
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    Cited by:

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    3. M’hamed Gaïgi & Idris Kharroubi & Thomas Lim, 2020. "Optimal Exploitation of a General Renewable Natural Resource under State and Delay Constraints," Mathematics, MDPI, vol. 8(11), pages 1-25, November.
    4. Fabbri Giorgio & Federico Salvatore, 2014. "On the Infinite-Dimensional Representation of Stochastic Controlled Systems with Delayed Control in the Diffusion Term," Mathematical Economics Letters, De Gruyter, vol. 2(3-4), pages 33-43, November.
    5. Luciano Campi & Davide Santis, 2020. "Nonzero-Sum Stochastic Differential Games Between an Impulse Controller and a Stopper," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 688-724, August.
    6. Dai, Min & Huang, Shan & Keppo, Jussi, 2019. "Opaque bank assets and optimal equity capital," Journal of Economic Dynamics and Control, Elsevier, vol. 100(C), pages 369-394.
    7. Álvaro Cartea & Leandro Sánchez-Betancourt, 2023. "Optimal execution with stochastic delay," Finance and Stochastics, Springer, vol. 27(1), pages 1-47, January.
    8. Magnus Perninge, 2018. "A limited-feedback approximation scheme for optimal switching problems with execution delays," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(3), pages 347-382, June.
    9. Frédéric Zumer & Jacques Le Cacheux & Marc Flandreau, 1998. "Stability without a pact? Lessons from the European Gold Standard, 1880-1913," Sciences Po publications n°98-01, Sciences Po.

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