Relationship between Maximum Principle and Dynamic Programming in Stochastic Differential Games and Applications

Jingtao Shi

School of Mathematics, Shandong University, Jinan, China.

DOI: 10.4236/ajor.2013.36043                

Abstract

This paper is concerned with the relationship between maximum principle and dynamic programming in zero-sum stochastic differential games. Under the assumption that the value function is enough smooth, relations among the adjoint processes, the generalized Hamiltonian function and the value function are given. A portfolio optimization problem under model uncertainty in the financial market is discussed to show the applications of our result.

Keywords

Stochastic Optimal Control; Stochastic Differential Games; Dynamic Programming; Maximum Principle; Portfolio Optimization; Model Uncertainty

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. J. Elliott, “The Existence of Value in Stochastic Differential Games,” SIAM Journal on Control and Optimization, Vol. 14, No. 1, 1976, pp. 85-94. http://dx.doi.org/10.1137/0314006
[2]	W. H. Fleming and P. E. Souganidis, “On the Existence of Value Function of Two-Player, Zero-Sum Stochastic Differential Games,” Indiana University Mathematics Journal, Vol. 38, No. 2, 1989, pp. 293-314. http://dx.doi.org/10.1512/iumj.1989.38.38015
[3]	R. Buckdahn, P. Cardaliaguet and C. Rainer, “Nash Equilibrium Payoffs for Nonzero-sum Stochastic Differential Games,” SIAM Journal on Control and Optimization, Vol. 43, No. 2, 2004, pp. 624-642. http://dx.doi.org/10.1137/S0363012902411556
[4]	S. Matatamvura and B. Oksendal, “Risk Minimizing Portfolios and HJBI Equations for Stochastic Differential Games,” Stochastics: An International Journal of Probability and Stochastics Processes, Vol. 80, No. 4, 2008, pp. 317-337. http://dx.doi.org/10.1080/17442500701655408
[5]	S. J. Tang and X. J. Li, “Differential Games of N Players in Stochastic Systems with Controlled Diffusion Terms,” Automatic and Remote Control, Vol. 69, No. 5, 2008, pp. 874-890. http://dx.doi.org/10.1134/S0005117908050123
[6]	T. T. K. An and B. Oksendal, “Maximum Principle for Stochastic Differential Games with Partial Information,” Journal of Optimization Theory and Applications, Vol. 139, No. 3, 2008, pp. 463-483. http://dx.doi.org/10.1007/s10957-008-9398-y
[7]	A. Bensoussan, “Maximum Principle and Dynamic Programming of the Optimal Control of Partially Observed Diffusions,” Stochastics, Vol. 9, No. 3, 1989, pp. 169-222. http://dx.doi.org/10.1080/17442508308833253
[8]	X. Y. Zhou, “The Connection between the Maximum Principle and Dynamic Programming in Stochastic Controll,” Stochastics & Stochastics Reports, Vol. 31, No. 1-4, 1990, pp. 1-13. http://dx.doi.org/10.1080/17442509008833645
[9]	J. M. Yong and X. Y. Zhou, “Stochastic Controls: Hamiltonian Systems and HJB Equations,” Springer-Verlag, New York, 1999. http://dx.doi.org/10.1007/978-1-4612-1466-3
[10]	N. C. Framstad, B. Oksendal and A. Sulem, “A Sufficient Stochastic Maximum Principle for Optimal Control of Jump Diffusions and Applications to Finance,” Journal of Optimization Theory and Applications, Vol. 121, No. 1, 2004, pp. 77-98. http://dx.doi.org/10.1023/B:JOTA.0000026132.62934.96
[11]	J. T. Shi and Z. Wu, “Relationship between MP and DPP for the Optimal Control Problem of Jump Diffusions,” Applied Mathematics and Optimization, Vol. 63, No. 2, 2011, pp. 151-189. http://dx.doi.org/10.1007/s00245-010-9115-8
[12]	C. Donnelly, “Sufficient Stochastic Maximum Principle in a Regime-Switching Diffusion Model,” Applied Mathematics and Optimization, Vol. 64, No. 2, 2011, pp. 155-169. http://dx.doi.org/10.1007/s00245-010-9130-9
[13]	G. Peskir and J. Shorish, “Market Forces and Dynamic Asset Pricing,” Stochastic Analysis and Applications, Vol. 20, No. 5, 2002, pp. 1027-1082. http://dx.doi.org/10.1081/SAP-120014553
[14]	R. Korn and O. Menkens, “Worst-Case Scenario Portfolio Optimization: A New Stochastic Control Approach,” Mathematical Methods of Operations Research, Vol. 62, No. 1, 2005, pp. 123-140. http://dx.doi.org/10.1007/s00186-005-0444-3
[15]	B. Oksendal and A. Sulem, “A Game Theoretical Approach to Martingale Measures in Incomplete Markets,” Pure Mathematics, E-Print of University of Oslo, No. 24, 2006.
[16]	B. Oksendal and A. Sulem, “Forward-Backward SDE Games and Stochastic Control under Model Uncertainty,” Journal of Optimization Theory and Applications, in Press.