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Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

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  • Fuhrman, Marco
  • Morlais, Marie-Amélie

Abstract

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:3120-3153
    DOI: 10.1016/j.spa.2019.09.008
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/5733 is not listed on IDEAS
    2. Bandini, Elena & Fuhrman, Marco, 2017. "Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1441-1474.
    3. Elie, Romuald & Kharroubi, Idris, 2010. "Probabilistic representation and approximation for coupled systems of variational inequalities," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1388-1396, September.
    4. 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.
    5. Carr, Peter, 1998. "Randomization and the American Put," Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 597-626.
    6. Dixit, Avinash K, 1989. "Entry and Exit Decisions under Uncertainty," Journal of Political Economy, University of Chicago Press, vol. 97(3), pages 620-638, June.
    7. Cosso, Andrea & Fuhrman, Marco & Pham, Huyên, 2016. "Long time asymptotics for fully nonlinear Bellman equations: A backward SDE approach," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1932-1973.
    8. Bandini, Elena & Cosso, Andrea & Fuhrman, Marco & Pham, Huyên, 2019. "Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 674-711.
    9. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    10. 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.
    11. Hamadène, Said & Zhang, Jianfeng, 2010. "Switching problem and related system of reflected backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 403-426, April.
    12. repec:dau:papers:123456789/12195 is not listed on IDEAS
    13. Brennan, Michael J & Schwartz, Eduardo S, 1985. "Evaluating Natural Resource Investments," The Journal of Business, University of Chicago Press, vol. 58(2), pages 135-157, April.
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