IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2302.08809.html
   My bibliography  Save this paper

Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic models

Author

Listed:
  • Filippo de Feo
  • Salvatore Federico
  • Andrzej 'Swik{e}ch

Abstract

In this manuscript we consider a class optimal control problem for stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a $C^{1,\alpha}$-partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising).

Suggested Citation

  • Filippo de Feo & Salvatore Federico & Andrzej 'Swik{e}ch, 2023. "Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic models," Papers 2302.08809, arXiv.org.
    • Handle: RePEc:arx:papers:2302.08809
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2302.08809
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Marina Di Giacinto & Salvatore Federico & Fausto Gozzi, 2011. "Pension funds with a minimum guarantee: a stochastic control approach," Finance and Stochastics, Springer, vol. 15(2), pages 297-342, June.
    2. Goldys, B. & Gozzi, F., 2006. "Second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces and stochastic control: approach," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1932-1963, December.
    3. Faggian, Silvia & Gozzi, Fausto, 2010. "Optimal investment models with vintage capital: Dynamic programming approach," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 416-437, July.
    4. 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.
    5. Salvatore Federico, 2011. "A stochastic control problem with delay arising in a pension fund model," Finance and Stochastics, Springer, vol. 15(3), pages 421-459, September.
    6. Mauro Bambi & Cristina Girolami & Salvatore Federico & Fausto Gozzi, 2017. "Generically distributed investments on flexible projects and endogenous growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(2), pages 521-558, February.
    7. Salvatore Federico & Paul Gassiat & Fausto Gozzi, 2015. "Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation," Finance and Stochastics, Springer, vol. 19(2), pages 415-448, April.
    8. Djehiche, Boualem & Gozzi, Fausto & Zanco, Giovanni & Zanella, Margherita, 2022. "Optimal portfolio choice with path dependent benchmarked labor income: A mean field model," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 48-85.
    9. F. Gozzi & C. Marinelli & S. Savin, 2009. "On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects," Journal of Optimization Theory and Applications, Springer, vol. 142(2), pages 291-321, August.
    10. Enrico Biffis & Fausto Gozzi & Cecilia Prosdocimi, 2020. "Optimal portfolio choice with path dependent labor income: the infinite horizon case," Papers 2002.00201, arXiv.org.
    11. 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.
    12. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    2. 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 1-11, November.
    3. 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.
    4. Enrico Biffis & Beniamin Goldys & Cecilia Prosdocimi & Margherita Zanella, 2023. "A pricing formula for delayed claims: appreciating the past to value the future," Mathematics and Financial Economics, Springer, volume 17, number 2, June.
    5. Mauro Bambi & Cristina Girolami & Salvatore Federico & Fausto Gozzi, 2017. "Generically distributed investments on flexible projects and endogenous growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(2), pages 521-558, February.
    6. d’Albis, Hippolyte & Augeraud-Véron, Emmanuelle & Hupkes, Hermen Jan, 2014. "Multiple solutions in systems of functional differential equations," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 50-56.
    7. Raouf Boucekkine & Giorgio Fabbri & Patrick-Antoine Pintus, 2011. "On the optimal control of a linear neutral differential equation arising in economics," Working Papers halshs-00576770, HAL.
    8. Fabbri, Giorgio, 2017. "International borrowing without commitment and informational lags: Choice under uncertainty," Journal of Mathematical Economics, Elsevier, vol. 68(C), pages 103-114.
    9. Marina Di Giacinto & Salvatore Federico & Fausto Gozzi, 2011. "Pension funds with a minimum guarantee: a stochastic control approach," Finance and Stochastics, Springer, vol. 15(2), pages 297-342, June.
    10. Faggian, Silvia & Gozzi, Fausto & Kort, Peter M., 2021. "Optimal investment with vintage capital: Equilibrium distributions," Journal of Mathematical Economics, Elsevier, vol. 96(C).
    11. Emmanuelle Augeraud-Veron & Mauro Bambi, 2012. "Does habit formation always increase the agents' desire to smooth consumption?," Discussion Papers 12/12, Department of Economics, University of York.
    12. BOUCEKKINE, Raouf & FABBRI, Giorgio & PINTUS, Patrick, 2012. "On the optimal control of a linear neutral differential equation arising in economics," LIDAM Reprints CORE 2449, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. Mauro Bambi & Cristina Di Girolami & Salvatore Federico & Fausto Gozzi, 2014. "On the Consequences of Generically Distributed Investments on Flexible Projects in an Endogenous Growth Model," Discussion Papers 14/15, Department of Economics, University of York.
    14. Cristiano Ricci, 2023. "A non-invariance result for the spatial AK model," Papers 2311.06811, arXiv.org.
    15. Emmanuelle Augeraud-Veron & Mauro Bambi & Fausto Gozzi, 2017. "Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 584-611, May.
    16. Raouf Boucekkine & Giorgio Fabbri & Fausto Gozzi, 2011. "Revisiting the Optimal Population Size Problem under Endogenous Growth: Minimal Utility Level and Finite Life," Asia-Pacific Journal of Accounting & Economics, Taylor & Francis Journals, vol. 18(3), pages 287-305.
    17. Yuki SHIGETA, 2022. "A Continuous-Time Utility Maximization Problem with Borrowing Constraints in Macroeconomic Heterogeneous Agent Models:A Case of Regular Controls under Markov Chain Uncertainty," Discussion papers e-22-009, Graduate School of Economics , Kyoto University.
    18. Emmanuelle Augeraud-Véron & Raouf Boucekkine & Vladimir Veliov, 2019. "Distributed Optimal Control Models in Environmental Economics: A Review," AMSE Working Papers 1902, Aix-Marseille School of Economics, France.
    19. Enrico Biffis & Beniamin Goldys & Cecilia Prosdocimi & Margherita Zanella, 2015. "A pricing formula for delayed claims: Appreciating the past to value the future," Papers 1505.04914, arXiv.org, revised Jul 2022.
    20. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Working Papers hal-03145949, HAL.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2302.08809. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.