Regression Monte Carlo for impulse control. (English) Zbl 1508.93157
Impulse control problems are difficult to be solved by the classical techniques using quasi-variational inequalities, because of the non-local term in the equation. The numerical algorithm in the present paper is of probabilistic-statistical type, inspired by Monte Carlo-techniques initially developed for stopping problems. It is based on the dynamic programming equation, which is discretized and solved backwards, the continuation function and the optimal impulse function being both statistically estimated through the simulation of state trajectories. Several implementations are discussed, and the algorithm is tested on two examples, treating forest rotation and irreversible investment. A package for R is publicly available on GitHub.
Reviewer: Catherine Rainer (Brest)
MSC:
| 93C27 | Impulsive control/observation systems |
| 65C05 | Monte Carlo methods |
| 93E20 | Optimal stochastic control |
| 49N25 | Impulsive optimal control problems |
References:
| [1] | Aid, René; Federico, Salvatore; Pham, Huyên; Villeneuve, Bertrand, Explicit investment rules with time-to-build and uncertainty, J. Econ. Dyn. Control, 51, 240-256 (2015) · Zbl 1402.91658 |
| [2] | Alvarez, Luis H. R., A class of solvable impulse control problems, Appl. Math. Optim., 49, 3, 265-295 (2004) · Zbl 1060.93105 · doi:10.1007/s00245-004-0792-z |
| [3] | Alvarez, Luis H. R., Optimal capital accumulation under price uncertainty and costly reversibility, J. Econ. Dyn. Control, 35, 10, 1769-1788 (2011) · Zbl 1282.91170 · doi:10.1016/j.jedc.2011.06.006 |
| [4] | Alvarez, Luis H. R.; Koskela, Erkki, Optimal harvesting under resource stock and price uncertainty, J. Econ. Dyn. Control, 31, 7, 2461-2485 (2007) · Zbl 1163.91509 · doi:10.1016/j.jedc.2006.08.003 |
| [5] | Alvarez, Luis H. R.; Koskela, Erkki, Taxation and rotation age under stochastic forest stand value, J. Environ. Econ. Manage., 54, 1, 113-127 (2007) · Zbl 1155.91433 · doi:10.1016/j.jeem.2006.11.002 |
| [6] | Alvarez, Luis H. R.; Lempa, Jukka, On the optimal stochastic impulse control of linear diffusions, SIAM J. Control Optim., 47, 2, 703-732 (2008) · Zbl 1157.49040 · doi:10.1137/060659375 |
| [7] | Azcue, Pablo; Muler, Nora; Palmowski, Zbigniew, Optimal dividend payments for a two-dimensional insurance risk process, Eur. Actuar. J., 9, 1, 241-272 (2019) · Zbl 1422.91324 · doi:10.1007/s13385-018-0182-6 |
| [8] | Azimzadeh, Parsiad; Bayraktar, Erhan; Labahn, George, Convergence of implicit schemes for Hamilton-Jacobi-Bellman quasi-Variational inequalities, SIAM J. Control Optim., 56, 6, 3994-4016 (2018) · Zbl 1405.49021 · doi:10.1137/18M1171965 |
| [9] | Basei, Matteo, Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates, Math. Methods Oper. Res., 89, 3, 355-383 (2019) · Zbl 1417.91209 · doi:10.1007/s00186-019-00665-x |
| [10] | Bayraktar, Erhan; Kyprianou, Andreas E.; Yamazaki, Kazutoshi, Optimal dividends in the dual model under transaction costs, Insur. Math. Econ., 54, 133-143 (2014) · Zbl 1294.91071 · doi:10.1016/j.insmatheco.2013.11.007 |
| [11] | Bayraktar, Erhan; Ludkovski, Michael, Inventory management with partially observed nonstationary demand, Ann. Oper. Res., 176, 1, 7-39 (2010) · Zbl 1194.90008 · doi:10.1007/s10479-009-0513-8 |
| [12] | Belak, Christoph; Christensen, Sören; Seifried, Frank Thomas, A general verification result for stochastic impulse control problems, SIAM J. Control Optim., 55, 2, 627-649 (2017) · Zbl 1358.93186 · doi:10.1137/16M1082822 |
| [13] | Bensoussan, Alain; Chevalier-Roignant, Benoît, Sequential capacity expansion options, Oper. Res., 67, 1, 33-57 (2019) · Zbl 1455.91268 · doi:10.1287/opre.2018.1752 |
| [14] | Bensoussan, Alain; Liu, R. H.; Sethi, Suresh P., Optimality of an \((s,{S})\) policy with compound Poisson and diffusion demands: A quasi-variational inequalities approach, SIAM J. Control Optim., 44, 5, 1650-1676 (2005) · Zbl 1151.90304 · doi:10.1137/S0363012904443737 |
| [15] | Cadenillas, Abel; Zapatero, Fernando, Classical and impulse stochastic control of the exchange rate using interest rates and reserves, Math. Finance, 10, 2, 141-156 (2000) · Zbl 1034.91036 · doi:10.1111/1467-9965.00086 |
| [16] | Chen, Yann-Shin Aaron; Guo, Xin, Impulse control of multidimensional jump diffusions in finite time horizon, SIAM J. Control Optim., 51, 3, 2638-2663 (2013) · Zbl 1275.49062 · doi:10.1137/110854205 |
| [17] | Christensen, Sören, On the solution of general impulse control problems using superharmonic functions, Stochastic Processes Appl., 124, 1, 709-729 (2014) · Zbl 1281.49033 · doi:10.1016/j.spa.2013.09.008 |
| [18] | Czarna, Irmina; Palmowski, Zbigniew, De Finetti’s dividend problem and impulse control for a two-dimensional insurance risk process, Stoch. Models, 27, 2, 220-250 (2011) · Zbl 1214.91051 · doi:10.1080/15326349.2011.567930 |
| [19] | Egami, Masahiko, A direct solution method for stochastic impulse control problems of one-dimensional diffusions, SIAM J. Control Optim., 47, 3, 1191-1218 (2008) · Zbl 1161.49036 · doi:10.1137/060669905 |
| [20] | Egloff, Daniel; Kohler, Michael; Todorovic, Nebojsa, A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options, Ann. Appl. Probab., 17, 4, 1138-1171 (2007) · Zbl 1136.91010 |
| [21] | El Asri, Brahim; Mazid, Sehail, Stochastic impulse control Problem with state and time dependent cost functions, Math. Control Relat. Fields, 10, 4, 855 (2020) · Zbl 1461.93541 · doi:10.3934/mcrf.2020022 |
| [22] | El Asri, Brahim; Mazid, Sehail, Zero-sum stochastic differential game in finite horizon involving impulse controls, Appl. Math. Optim., 81, 3, 1055-1087 (2020) · Zbl 1454.91020 · doi:10.1007/s00245-018-9529-2 |
| [23] | Federico, Salvatore; Rosestolato, Mauro; Tacconi, Elisa, Irreversible investment with fixed adjustment costs: a stochastic impulse control approach, Math. Financ. Econ., 13, 4, 579-616 (2019) · Zbl 1422.91649 · doi:10.1007/s11579-019-00238-w |
| [24] | Guthrie, Graeme, Uncertainty and the trade-off between scale and flexibility in investment, J. Econ. Dyn. Control, 36, 11, 1718-1728 (2012) · Zbl 1345.91016 · doi:10.1016/j.jedc.2012.04.008 |
| [25] | Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome, The elements of statistical learning: data mining, inference and prediction (2009), Springer · Zbl 1273.62005 · doi:10.1007/978-0-387-84858-7 |
| [26] | Hu, Jianqiang; Zhang, Cheng; Zhu, Chenbo, \((s, {S})\) inventory systems with correlated demands, INFORMS J. Comput., 28, 4, 603-611 (2016) · Zbl 1357.90006 |
| [27] | Kohler, Michael, A regression-based smoothing spline Monte Carlo algorithm for pricing American options in discrete time, AStA, Adv. Stat. Anal., 92, 2, 153-178 (2008) · Zbl 1477.62295 · doi:10.1007/s10182-008-0067-0 |
| [28] | Longstaff, Francis A.; Schwartz, Eduardo S., Valuing American options by simulations: a simple least squares approach, Rev. Financ. Stud., 14, 113-148 (2001) · Zbl 1386.91144 · doi:10.1093/rfs/14.1.113 |
| [29] | Ludkovski, Mike, mlOSP: Towards a Unified Implementation of Regression Monte Carlo Algorithms (2020) |
| [30] | Øksendal, Bernt Karsten; Sulem, Agnes, Applied stochastic control of jump diffusions, 498 (2007), Springer · Zbl 1116.93004 · doi:10.1007/978-3-540-69826-5 |
| [31] | Tsitsiklis, John; Van Roy, Benjamin, Regression Methods for Pricing Complex American-Style Options, IEEE Trans. Neural Netw., 12, 4, 694-703 (2001) · doi:10.1109/72.935083 |
| [32] | Williams, Christopher K. I.; Rasmussen, Carl Edward, Gaussian processes for machine learning (2006), MIT Press · Zbl 1177.68165 |
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