Abstract
This paper presents a sufficient stochastic maximum principle for a stochastic optimal control problem of Markov regime-switching forward–backward stochastic differential equations with jumps. The relationship between the stochastic maximum principle and the dynamic programming principle in a Markovian case is also established. Finally, applications of the main results to a recursive utility portfolio optimization problem in a financial market are discussed.
Similar content being viewed by others
References
Donnelly, C.: Sufficient stochastic maximum principle in a regime-switching diffusion model. Appl. Math. Optim. 64(2), 155–169 (2011)
Donnelly, C., Heunis, A.J.: Quadratic risk minimization in a regime-switching model with portfolio constraints. SIAM J. Control Optim. 50(4), 2431–2461 (2012)
Tao, R., Wu, Z.: Maximum principle for optimal control problems of forward–backward regime-switching system and applications. Syst. Control Lett. 61(9), 911–917 (2012)
Zhang, X., Elliott, R.J., Siu, T.K.: A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance. SIAM J. Control Optim. 50(2), 964–990 (2012)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006)
Peng, S.: A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28(4), 966–979 (1990)
Tang, S., Li, X.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32(5), 1447–1475 (1994)
Framstad, N.C., Øksendal, B., Sulem, A.: Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance. J. Optim. Theory Appl. 121(1), 77–98 (2004)
Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)
Bismut, J.M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44(2), 384–404 (1973)
Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)
Duffie, D., Epstein, L.G.: Stochastic differential utility. Econom. J. Econom. Soc. 60, 353–394 (1992)
Karoui, N.E., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Financ. 7(1), 1–71 (1997)
Peng, S.: Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27(2), 125–144 (1993)
Shi, J., Wu, Z.: Maximum principle for forward–backward stochastic control system with random jumps and applications to finance. J. Syst. Sci. Complex. 23(2), 219–231 (2010)
Øksendal, B., Sulem, A.: Maximum principles for optimal control of forward–backward stochastic differential equations with jumps. SIAM J. Control Optim. 48(5), 2945–2976 (2009)
Pamen, O.M.: Maximum principles of Markov regime-switching forward–backward stochastic differential equations with jumps and partial information. arXiv:1403.2901 (2014)
Peng, S.: A generalized dynamic programming principle and Hamilton–Jacobi–Bellman equation. Stochastics 38(2), 119–134 (1992)
Peng, S.: Backward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJB equations. In: Topics on Stochastic Analysis. pp. 85–138, Science Press, Beijing, China (1997)
Li, J., Peng, S.: Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of HJB equations. Nonlinear Anal. Theory Methods Appl. 70(4), 1776–1796 (2009)
Shi, J., Yu, Z.: Relationship between maximum principle and dynamic programming for stochastic recursive optimal control problems and applications. Math. Probl. Eng. 2013, 1–12 (2013)
Elliott, R.J., Aggoun, L., Moore, J.B.: Hidden Markov Models: Estimation and Control. Springer, New York (1994)
Cohen, S.N., Elliott, R.J.: Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions. Ann. Appl. Probab. 20(1), 267–311 (2010)
Shi, J., Wu, Z.: Relationship between MP and DPP for the stochastic optimal control problem of jump diffusions. Appl. Math. Optim. 63(2), 151–189 (2011)
Shi, J.: Relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problems of jump diffusions. Optim. Control Appl. Methods 35(1), 61–76 (2014)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, New York (1992)
Acknowledgements
The authors would like to thank the referees for their careful reading of the paper and helpful suggestions. This work is supported by the National Natural Science Foundation of China (NSFC Grant Nos. 11571189, 11371020) and the European Union’s Seventh Framework Programme for research, technological development and demonstration under Grant Agreement No. 318984-RARE.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Christiane Tammer.
Rights and permissions
About this article
Cite this article
Sun, Z., Guo, J. & Zhang, X. Maximum Principle for Markov Regime-Switching Forward–Backward Stochastic Control System with Jumps and Relation to Dynamic Programming. J Optim Theory Appl 176, 319–350 (2018). https://doi.org/10.1007/s10957-017-1068-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-017-1068-5
Keywords
- Stochastic maximum principle
- Regime-switching
- Forward–backward stochastic differential equations
- Dynamic programming
- Recursive utility optimization