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A change of variable formula with applications to multi-dimensional optimal stopping problems

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  • Cai, Cheng
  • De Angelis, Tiziano

Abstract

We derive a change of variable formula for C1 functions U:R+×Rm→R whose second order spatial derivatives may explode and not be integrable in the neighbourhood of a surface b:R+×Rm−1→R that splits the state space into two sets C and D. The formula is tailored for applications in problems of optimal stopping where it is generally very hard to control the second order derivatives of the value function near the optimal stopping boundary. Differently to other existing papers on similar topics we only require that the surface b be monotonic in each variable and we formally obtain the same expression as the classical Itô’s formula.

Suggested Citation

  • Cai, Cheng & De Angelis, Tiziano, 2023. "A change of variable formula with applications to multi-dimensional optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 33-61.
    • Handle: RePEc:eee:spapps:v:164:y:2023:i:c:p:33-61
    DOI: 10.1016/j.spa.2023.07.005
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    References listed on IDEAS

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    1. Goran Peskir, 2005. "A Change-of-Variable Formula with Local Time on Curves," Journal of Theoretical Probability, Springer, vol. 18(3), pages 499-535, July.
    2. Tiziano De Angelis & Erik Ekstrom, 2016. "The dividend problem with a finite horizon," Papers 1609.01655, arXiv.org, revised Nov 2017.
    3. Elena Bandini & Tiziano De Angelis & Giorgio Ferrari & Fausto Gozzi, 2022. "Optimal dividend payout under stochastic discounting," Mathematical Finance, Wiley Blackwell, vol. 32(2), pages 627-677, April.
    4. Rozkosz, Andrzej, 1996. "Stochastic representation of diffusions corresponding to divergence form operators," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 11-33, October.
    5. Tiziano De Angelis & Salvatore Federico & Giorgio Ferrari, 2014. "Optimal Boundary Surface for Irreversible Investment with Stochastic Costs," Papers 1406.4297, arXiv.org, revised Jan 2017.
    6. Tiziano De Angelis & Salvatore Federico & Giorgio Ferrari, 2017. "Optimal Boundary Surface for Irreversible Investment with Stochastic Costs," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1135-1161, November.
    7. Pavel V. Gapeev, 2016. "Bayesian Switching Multiple Disorder Problems," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 1108-1124, August.
    8. Christensen, Sören & Crocce, Fabián & Mordecki, Ernesto & Salminen, Paavo, 2019. "On optimal stopping of multidimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2561-2581.
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    Cited by:

    1. De Angelis, Tiziano & Gensbittel, Fabien & Villeneuve, Stéphane, 2023. "Nash equilibria for dividend distribution with competition," TSE Working Papers 23-1495, Toulouse School of Economics (TSE), revised Jul 2025.
    2. Filippo de Feo, 2025. "Stochastic optimal control problems with measurable coefficients via $L^p$-viscosity solutions and applications to optimal advertising models," Papers 2502.02352, arXiv.org.
    3. Tiziano de Angelis & Fabien Gensbittel & Stéphane Villeneuve, 2025. "Nash Equilibria for Dividend Distribution with Competition," Post-Print hal-05345639, HAL.
    4. Federico, Salvatore & Ferrari, Giorgio & Torrente, Maria Laura, 2023. "Irreversible Reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost," Center for Mathematical Economics Working Papers 682, Center for Mathematical Economics, Bielefeld University.
    5. Salvatore Federico & Giorgio Ferrari & Maria-Laura Torrente, 2024. "Irreversible reinsurance: minimization of capital injections in presence of a fixed cost," Mathematics and Financial Economics, Springer, volume 18, number 6, November.

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