×

Finite horizon impulse control of stochastic functional differential equations. (English) Zbl 1515.60197

Summary: In this work we show that one can solve a finite horizon non-Markovian impulse control problem with control dependent dynamics. This dynamic satisfies certain functional Lipschitz conditions and is path dependent in such a way that the resulting trajectory becomes a flow.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G40 Stopping times; optimal stopping problems; gambling theory
93E20 Optimal stochastic control
62P20 Applications of statistics to economics
91B99 Mathematical economics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bar-Ilan, A. and Sulem, A., Explicit solution of inventory problems with delivery lags, Math. Oper. Res., 20 (1995). · Zbl 0846.90031
[2] Belak, C., Christensen, S., and Seifried, F. T., A general verification result for stochastic impulse control problems, SIAM J. Control Optim., 55 (2017), pp. 627-649. · Zbl 1358.93186
[3] Bensoussan, A. and Lions, J., Impulse Control and Quasivariational inequalities, Gauthier-Villars, Montrouge, 1984.
[4] Brekke, K. A. and Øksendal, B., The high contact principle as a sufficiency condition for optimal stopping, in Stochastic Models and Option Values: Applications to Resources, Environment, and Investment Problems, , North-Holland, Amsterdam, 1991. · Zbl 0783.90019
[5] Bruder, B. and Pham, H., Impulse control problem on finite horizon with execution delay, Stochastic Process. Appl., 119 (2009), pp. 1436-1469. · Zbl 1159.93361
[6] Djehiche, B., Hamadène, S., and Hdhiri, I., Stochastic impulse control of non-Markovian processes, Appl. Math. Optim., 61 (2010), pp. 1-26. · Zbl 1195.93144
[7] Djehiche, B., Hamadène, S., and Popier, A., A finite horizon optimal multiple switching problem, SIAM J. Control Optim., 48 (2009), pp. 2751-2770. · Zbl 1196.60069
[8] Dynkin, E. B., The optimum choice of the instant for stopping a Markov process, Soviet Math. Dokl., 4 (1963), pp. 627-629. · Zbl 0242.60018
[9] Karoui, N. El, Les aspects probabilistes du controle stochastique, in Proceedings of Ecole d’Eté de Probabilistés de Saint-Flour IX—1979, , Springer, New York, 1981. · Zbl 0472.60002
[10] Karoui, N. El, Kapoudjian, C., Pardoux, E., Peng, S., and Quenez, M. C., Reflected solutions of backward SDEs and related obstacle problems for PDEs, Ann. Probab., 25 (1997), pp. 702-737. · Zbl 0899.60047
[11] Karoui, N. El and Tan, X., Capacities, Measurable Selection and Dynamic Programming. I, preprint, 2013.
[12] Hamadène, S. and Jeanblanc, M., On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), pp. 182-192. · Zbl 1276.91100
[13] Huré, C., Pham, H., Bachouch, A., and Langrené, N., Deep neural networks algorithms for stochastic control problems on finite horizon: Convergence analysis, SIAM J. Numer. Anal., 59 (2021), pp. 525-557. · Zbl 1466.65007
[14] Kobylanski, M. and Quenez, M. C., Optimal stopping in a general framework, Electron. J. Probab., 17 (2012), pp. 721-28. · Zbl 1405.60055
[15] Kushner, H. J., Numerical Methods for Controlled Stochastic Delay Systems, Birkhäuser Boston, Boston, MA, 2008. · Zbl 1219.93001
[16] Perninge, M., A finite horizon optimal switching problem with memory and application to controlled SDDEs, Math. Methods Oper. Res., 91 (2020), pp. 465-500. · Zbl 1448.49040
[17] Perninge, M., Infinite horizon impulse control of stochastic functional differential equations driven by Lévy processes, preprint, 2020.
[18] Peskir, G. and Shiryaev, A., Optimal Stopping and Free-Boundary Problems, Birkhäuser Verlag, Basel, 2006. · Zbl 1115.60001
[19] Aïd, R., Federico, S., Pham, H., and Villeneuve, B., Explicit investment rules with time-to-build and uncertainty, J. Econom. Dynam. Control, 51 (2015), pp. 240-256. · Zbl 1402.91658
[20] Protter, P., Stochastic Integration and Differential Equations, 2nd ed., Springer, New York, 2004. · Zbl 1041.60005
[21] Snell, J. L., Applications of martingale system theorems, Trans. Amer. Math. Soc., 73 (1952), pp. 293-312. · Zbl 0048.11402
[22] Wald, A., Sequential Analysis, Wiley, New York, 1947. · Zbl 0029.15805
[23] Øksendal, B. and Sulem, A., Applied Stochastic Control of Jump Diffusions, 2nd ed., Springer, New York, 2006. · Zbl 1074.93009
[24] Øksendal, B. and Sulem, A., Optimal stochastic impulse control with delayed reaction, Appl. Math. Optim., 58 (2008), pp. 243-255. · Zbl 1161.93029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.