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Singular stochastic control in the presence of a state-dependent yield structure

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  • Alvarez, Luis H. R.

Abstract

We consider the determination of the optimal singular stochastic control for maximizing the expected cumulative revenue flows in the presence of a state-dependent marginal yield measuring the instantaneous returns accrued from irreversibly exerting the singular policy. As in standard models of singular stochastic control, the underlying stochastic process is assumed to evolve according to a regular linear diffusion. We derive the value of the optimal strategy by relying on a combination of stochastic calculus, the classical theory of diffusions, and non-linear programming. We state a set of usually satisfied conditions under which the optimal policy is to reflect the controlled process downwards at an optimal threshold satisfying an ordinary first-order necessary condition for an optimum. We also consider the comparative static properties of the value and state a set of sufficient conditions under which it is concave. As a consequence, we are able to state a set of sufficient conditions under which the sign of the relationship between the volatility of the process and the value is negative.

Suggested Citation

  • Alvarez, Luis H. R., 2000. "Singular stochastic control in the presence of a state-dependent yield structure," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 323-343, April.
    • Handle: RePEc:eee:spapps:v:86:y:2000:i:2:p:323-343
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    Citations

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    Cited by:

    1. GAHUNGU, Joachim & SMEERS, Yves, 2011. "A real options model for electricity capacity expansion," LIDAM Discussion Papers CORE 2011044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Zhuo Jin & George Yin & Chao Zhu, 2011. "Numerical Solutions of Optimal Risk Control and Dividend Optimization Policies under A Generalized Singular Control Formulation," Papers 1111.2584, arXiv.org.
    3. Salvatore Federico & Giorgio Ferrari & Patrick Schuhmann, 2019. "A Model for the Optimal Management of Inflation," Department of Economics University of Siena 812, Department of Economics, University of Siena.
    4. Nuno M. Brites, 2022. "Optimal Harvesting of Stochastically Fluctuating Populations Driven by a Generalized Logistic SDE Growth Model," Mathematics, MDPI, vol. 10(17), pages 1-15, August.
    5. Alvarez E., Luis H.R. & Hening, Alexandru, 2022. "Optimal sustainable harvesting of populations in random environments," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 678-698.
    6. Federico, Salvatore & Ferrari, Giorgio & Schuhmann, Patrick, 2019. "A Model for the Optimal Management of Inflation," Center for Mathematical Economics Working Papers 624, Center for Mathematical Economics, Bielefeld University.
    7. Ferrari, Giorgio & Koch, Torben, 2018. "An optimal extraction problem with price impact," Center for Mathematical Economics Working Papers 603, Center for Mathematical Economics, Bielefeld University.
    8. Joachim Gahungu and Yves Smeers, 2012. "A Real Options Model for Electricity Capacity Expansion," RSCAS Working Papers 2012/08, European University Institute.
    9. Ferrari, Giorgio, 2018. "On a Class of Singular Stochastic Control Problems for Reflected Diffusions," Center for Mathematical Economics Working Papers 592, Center for Mathematical Economics, Bielefeld University.
    10. de Angelis, Tiziano & Ferrari, Giorgio, 2016. "Stochastic nonzero-sum games: a new connection between singular control and optimal stopping," Center for Mathematical Economics Working Papers 565, Center for Mathematical Economics, Bielefeld University.
    11. Pui Chan Lon & Mihail Zervos, 2011. "A Model for Optimally Advertising and Launching a Product," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 363-376, May.
    12. Giorgio Ferrari & Torben Koch, 2018. "An Optimal Extraction Problem with Price Impact," Papers 1812.01270, arXiv.org.
    13. Ky Q. Tran & Bich T. N. Le & George Yin, 2022. "Harvesting of a Stochastic Population Under a Mixed Regular-Singular Control Formulation," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 1106-1132, December.

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