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Time discretization and quantization methods for optimal multiple switching problem

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  • Gassiat, Paul
  • Kharroubi, Idris
  • Pham, Huyên

Abstract

In this paper, we study probabilistic numerical methods based on optimal quantization algorithms for computing the solution to optimal multiple switching problems with regime-dependent state process. We first consider a discrete-time approximation of the optimal switching problem, and analyse its rate of convergence. Given a time step h, the error is in general of order (hlog(1/h))1/2, and of order h1/2 when the switching costs do not depend on the state process. We next propose quantization numerical schemes for the space discretization of the discrete-time Euler state process. A Markovian quantization approach relying on the optimal quantization of the normal distribution arising in the Euler scheme is analysed. In the particular case of uncontrolled state process, we describe an alternative marginal quantization method, which extends the recursive algorithm for optimal stopping problems as in Bally (2003) [1]. A priori Lp-error estimates are stated in terms of quantization errors. Finally, some numerical tests are performed for an optimal switching problem with two regimes.

Suggested Citation

  • Gassiat, Paul & Kharroubi, Idris & Pham, Huyên, 2012. "Time discretization and quantization methods for optimal multiple switching problem," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2019-2052.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:5:p:2019-2052
    DOI: 10.1016/j.spa.2012.02.008
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    References listed on IDEAS

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    1. Said Hamadène & Monique Jeanblanc, 2007. "On the Starting and Stopping Problem: Application in Reversible Investments," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 182-192, February.
    2. Rene Carmona & Michael Ludkovski, 2008. "Pricing Asset Scheduling Flexibility using Optimal Switching," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(5-6), pages 405-447.
    3. Bally, Vlad & Pagès, Gilles, 2003. "Error analysis of the optimal quantization algorithm for obstacle problems," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 1-40, July.
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    Cited by:

    1. Mihail Zervos & Carlos Oliveira & Kate Duckworth, 2018. "An investment model with switching costs and the option to abandon," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 417-443, December.
    2. Fuhrman, Marco & Morlais, Marie-Amélie, 2020. "Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3120-3153.
    3. Qinghua Li, 2014. "Facilitation and Internalization Optimal Strategy in a Multilateral Trading Context," Papers 1404.7320, arXiv.org, revised Jan 2015.
    4. Li Kai & Nyström Kaj & Olofsson Marcus, 2015. "Optimal switching problems under partial information," Monte Carlo Methods and Applications, De Gruyter, vol. 21(2), pages 91-120, June.
    5. Cortazar, Gonzalo & Naranjo, Lorenzo & Sainz, Felipe, 2021. "Optimal decision policy for real options under general Markovian dynamics," European Journal of Operational Research, Elsevier, vol. 288(2), pages 634-647.

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