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On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term

Author

Listed:
  • Giorgio Fabbri

    () (EPEE - Centre d'Etudes des Politiques Economiques - UEVE - Université d'Évry-Val-d'Essonne)

  • Salvatore Federico

    () (Dipartimento di economia, management e metodi quantitativi - Dipartimento di economia, management e metodi quantitativi - Università degli studi di Milano [Milano])

Abstract

In the deterministic context a series of well established results allow to reformulate delay differential equations (DDEs) as evolution equations in infinite dimensional spaces. Several models in the theoretical economic literature have been studied using this reformulation. On the other hand, in the stochastic case only few results of this kind are available and only for specific problems. The contribution of the present letter is to present a way to reformulate in infinite dimension a prototype controlled stochastic DDE, where the control variable appears delayed in the diffusion term. As application, we present a model for quadratic risk minimization hedging of European options with execution delay and a time-to-build model with shock. Some comments concerning the possible employment of the dynamic programming after the reformulation in infinite dimension conclude the letter.

Suggested Citation

  • Giorgio Fabbri & Salvatore Federico, 2014. "On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term," Post-Print hal-01038088, HAL.
  • Handle: RePEc:hal:journl:hal-01038088
    Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-01038088
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    References listed on IDEAS

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    1. Asea, Patrick K. & Zak, Paul J., 1999. "Time-to-build and cycles," Journal of Economic Dynamics and Control, Elsevier, vol. 23(8), pages 1155-1175, August.
    2. Raouf Boucekkine & David de la Croix & Omar Licandro, 2006. "Vintage Capital," Economics Working Papers ECO2006/8, European University Institute.
    3. Tsoukalas, John D., 2011. "Time to build capital: Revisiting investment-cash-flow sensitivities," Journal of Economic Dynamics and Control, Elsevier, vol. 35(7), pages 1000-1016, July.
    4. Boucekkine, Raouf & Licandro, Omar & Puch, Luis A. & del Rio, Fernando, 2005. "Vintage capital and the dynamics of the AK model," Journal of Economic Theory, Elsevier, vol. 120(1), pages 39-72, January.
    5. M. Bambi & G. Fabbri & F. Gozzi, 2012. "Optimal policy and consumption smoothing effects in the time-to-build AK model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(3), pages 635-669, August.
    6. d’Albis, Hippolyte & Augeraud-Veron, Emmanuelle & Venditti, Alain, 2012. "Business cycle fluctuations and learning-by-doing externalities in a one-sector model," Journal of Mathematical Economics, Elsevier, vol. 48(5), pages 295-308.
    7. Gustav Feichtinger & Richard F. Hartl & Suresh P. Sethi, 1994. "Dynamic Optimal Control Models in Advertising: Recent Developments," Management Science, INFORMS, vol. 40(2), pages 195-226, February.
    8. Mauro Bambi, 2006. "Endogenous Growth and Time-to-Build: the AK Case," Economics Working Papers ECO2006/17, European University Institute.
    9. Fabbri, Giorgio & Gozzi, Fausto, 2008. "Solving optimal growth models with vintage capital: The dynamic programming approach," Journal of Economic Theory, Elsevier, vol. 143(1), pages 331-373, November.
    10. 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.
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    Citations

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    Cited by:

    1. Fabbri, Giorgio, 2017. "International borrowing without commitment and informational lags: Choice under uncertainty," Journal of Mathematical Economics, Elsevier, vol. 68(C), pages 103-114.
    2. René Carmona & Jean-Pierre Fouque & Seyyed Mostafa Mousavi & Li-Hsien Sun, 0. "Systemic Risk and Stochastic Games with Delay," Journal of Optimization Theory and Applications, Springer, vol. 0, pages 1-34.
    3. René Carmona & Jean-Pierre Fouque & Seyyed Mostafa Mousavi & Li-Hsien Sun, 2018. "Systemic Risk and Stochastic Games with Delay," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 366-399, November.

    More about this item

    Keywords

    Stochastic delay differential equations; Evolution equations in Hilbert space; Dynamic programming;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • O40 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - General

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