## Abstract and Applied Analysis

### The Optimal Control Problem with State Constraints for Fully Coupled Forward-Backward Stochastic Systems with Jumps

Qingmeng Wei

#### Abstract

We focus on the fully coupled forward-backward stochastic differential equations with jumps and investigate the associated stochastic optimal control problem (with the nonconvex control and the convex state constraint) along with stochastic maximum principle. To derive the necessary condition (i.e., stochastic maximum principle) for the optimal control, first we transform the fully coupled forward-backward stochastic control system into a fully coupled backward one; then, by using the terminal perturbation method, we obtain the stochastic maximum principle. Finally, we study a linear quadratic model.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 216053, 12 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412606761

Digital Object Identifier
doi:10.1155/2014/216053

Mathematical Reviews number (MathSciNet)
MR3170392

#### Citation

Wei, Qingmeng. The Optimal Control Problem with State Constraints for Fully Coupled Forward-Backward Stochastic Systems with Jumps. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 216053, 12 pages. doi:10.1155/2014/216053. https://projecteuclid.org/euclid.aaa/1412606761

#### References

• É. Pardoux and S. G. Peng, “Adapted solution of a backward stochastic differential equation,” Systems & Control Letters, vol. 14, no. 1-2, pp. 55–61, 1990.
• H. Kushner, “Necessary conditions for continuous parameter stochastic optimization problems,” SIAM Journal on Control and Optimization, vol. 10, pp. 550–565, 1972.
• J.-M. Bismut, “An introductory approach to duality in optimal stochastic control,” SIAM Journal on Control and Optimization, vol. 20, no. 1, pp. 62–78, 1978.
• A. Bensoussan, “Lectures on stochastic control,” in Nonlinear Filtering and Stochastic Control, Lecture Notes in Mathematics, part I, pp. 1–39, Springer, New York, NY, USA, 1983.
• U. Haussmann, A Stochastic Maximum Principle for Optimal Control of Diffusions, vol. 151 of Pitman Research Notes in Mathematics, Wiley John & Sons, New York, NY, USA, 1986.
• Y. Hu, “Maximum principle of optimal control for Markov processes,” Acta Mathematica Sinica, vol. 33, pp. 43–56, 1990.
• S. G. Peng, “A general stochastic maximum principle for optimal control problems,” SIAM Journal on Control and Optimization, vol. 28, no. 4, pp. 966–979, 1990.
• S. G. Peng, “Backward stochastic differential equations and applications to optimal control,” Applied Mathematics and Optimization, vol. 27, no. 4, pp. 125–144, 1993.
• B. ${\text{\O}}$ksendal and A. Sulem, “Maximum principles for optimal control of forward-backward stochastic differential equations with jumps,” SIAM Journal on Control and Optimization, vol. 48, no. 5, pp. 2945–2976, 2009.
• J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43 of Applications of Mathematics, Springer, New York, NY, USA, 1999.
• W. S. Xu, “Stochastic maximum principle for optimal control problem of forward and backward system,” Journal of the Australian Mathematical Society B, vol. 37, no. 2, pp. 172–185, 1995.
• N. El Karoui, S. Peng, and M. C. Quenez, “A dynamic maximum principle for the optimization of recursive utilities under constraints,” The Annals of Applied Probability, vol. 11, no. 3, pp. 664–693, 2001.
• S. L. Ji and S. G. Peng, “Terminal perturbation method for the backward approach to continuous time mean-variance portfolio selection,” Stochastic Processes and their Applications, vol. 118, no. 6, pp. 952–967, 2008.
• S. L. Ji and X. Y. Zhou, “A maximum principle for stochastic optimal control with terminal state constraints, and its applications,” Communications in Information and Systems, vol. 6, no. 4, pp. 321–337, 2006.
• S. L. Ji and Z. Wu, “The maximum principle for one kind of stochastic optimization problem and application in dynamic measure of risk,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2189–2204, 2007.
• S. L. Ji and X. Y. Zhou, “The Neyman-Pearson lemma under $g$-probability,” Comptes Rendus de l'Académie des Sciences, vol. 346, no. 3-4, pp. 209–212, 2008.
• S. L. Ji and X. Y. Zhou, “A generalized Neyman-Pearson lemma under $g$-probabilities,” Probability Theory and Related Fields, vol. 148, no. 3-4, pp. 645–669, 2010.
• S. L. Ji, “Dual method for continuous-time Markowitz's problems with nonlinear wealth equations,” Journal of Mathematical Analysis and Applications, vol. 366, no. 1, pp. 90–100, 2010.
• J. T. Shi, “Necessary conditions for optimal control of forward-backward stochastic systems with random jumps,” International Journal of Stochastic Analysis, vol. 2012, Article ID 258674, 50 pages, 2012.
• J. T. Shi and Z. Wu, “Necessary condition for optimal control of fully coupled forward-backward stochastic system with random jumps,” in Proceedings of the 31th Chinese Control Conference, pp. 1620–1627, Hefei, China, 2012.
• R. Situ, “A maximum principle for optimal controls of stochastic with random jumps,” in Proceedings of the National Conference on Control Theory and Its Applications, Qingdao, China, October 1991.
• S. J. Tang and X. J. Li, “Necessary conditions for optimal control of stochastic systems with random jumps,” SIAM Journal on Control and Optimization, vol. 32, no. 5, pp. 1447–1475, 1999.
• J. T. Shi and Z. Wu, “Maximum principle for forward-backward stochastic control system with random jumps and applications to finance,” Journal of Systems Science & Complexity, vol. 23, no. 2, pp. 219–231, 2010.
• T. Bielecki, H. Jin, S. Pliska, and X. Y. Zhou, “Continuous-time mean-variance portfolio selection with bankruptcy prohibition,” Mathematical Finance, vol. 15, no. 2, pp. 213–244, 2005.
• C. Bernard, S. L. Ji, and W. D. Tian, “An optimal insurance design problem under Knightian uncertainty,” Decisions in Economics and Finance, vol. 36, no. 2, pp. 99–124, 2013.
• S. L. Ji and Q. M. Wei, “An overview on the principal-agent problems in continuous time,” in Real Options, Ambiguity, Risk and Insurance, A. Bensoussan, S. Peng, and J. Sung, Eds., Studies in Probability, Optimization and Statistics 5, IOS Press, Amsterdam, The Netherlands, 2013.
• S. L. Ji and Q. M. Wei, “A maximum principle for fully coupled forward-backward stochastic control systems with terminal state constraints,” Journal of Mathematical Analysis and Applications, vol. 407, no. 2, pp. 200–210, 2013.
• S. L. Ji, Q. M. Wei, and X. M. Zhang, “A maximum principle for controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal sate constraints,” Abstract and Applied Analysis, vol. 2012, Article ID 537376, 29 pages, 2012.
• L. Epstein and S. L. Ji, “Ambiguous volatility and asset pricing in continuous time,” The Review of Financial Studies, vol. 26, no. 7, pp. 1740–1786, 2013.
• L. Epstein and S. L. Ji, “Ambiguous volatility, possibility and utility in continuous time,” Journal of Mathematical Economics, 2013.
• Z. Wu, “Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration,” Journal of the Australian Mathematical Society, vol. 74, no. 2, pp. 249–266, 2003.
• G. Barles, R. Buckdahn, and E. Pardoux, “Backward stochastic differential equations and integral-partial differential equations,” Stochastics and Stochastics Reports, vol. 60, pp. 57–83, 1997.
• I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, pp. 324–353, 1974. \endinput