A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

doi:10.1016/S0252-9602(11)60242-7

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (2): 419-433.doi: 10.1016/S0252-9602(11)60242-7

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

史敬涛|吴臻

School of Mathematics, Shandong University, Jinan 250100, China

收稿日期:2008-02-25 修回日期:2008-10-05 出版日期:2011-03-20 发布日期:2011-03-20
基金资助:
This work was supported by the National Basic Research Program of China (973 Program, 2007CB814904), the National Natural Science Foundations of China (10921101) and Shandong Province (2008BS01024, ZR2010AQ004), the Science Funds for Distinguished Young Scholars of Shandong Province (JQ200801) and Shandong University (2009JQ004), and the Independent Innovation Foundations of Shandong University (IIFSDU, 2009TS036, 2010TS060)

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

SHI Jing-Tao, WU Zhen

School of Mathematics, Shandong University, Jinan 250100, China

Received:2008-02-25 Revised:2008-10-05 Online:2011-03-20 Published:2011-03-20
Supported by:
This work was supported by the National Basic Research Program of China (973 Program, 2007CB814904), the National Natural Science Foundations of China (10921101) and Shandong Province (2008BS01024, ZR2010AQ004), the Science Funds for Distinguished Young Scholars of Shandong Province (JQ200801) and Shandong University (2009JQ004), and the Independent Innovation Foundations of Shandong University (IIFSDU, 2009TS036, 2010TS060)

摘要/Abstract

摘要：

A stochastic maximum principle for the risk-sensitive optimal control problem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where the diffusion and jump term may both depend on the control. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equations and the maximum condition heavily depend on the risk-sensitive parameter. As applications, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and explicit optimal control is obtained.

关键词: Risk-sensitive control, jump diffusions, maximum principle, adjoint equation

Abstract:

Key words: Risk-sensitive control, jump diffusions, maximum principle, adjoint equation

中图分类号:

93E20

史敬涛, 吴臻. A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS[J]. 数学物理学报(英文版), 2011, 31(2): 419-433.

SHI Jing-Tao, WU Zhen. A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS[J]. Acta mathematica scientia,Series B, 2011, 31(2): 419-433.

参考文献

[1] Charalambous C D, Hibey J L. Minimum principle for partially observable nonlinear risk-sensitive control problems using measure-valued decompositions. Stoch Stoch Reports, 1996, 57: 247--288

[2] Cont R, Tankov P. Financial modelling with jump processes. New York: Chapmam Hall/CRC, 2004

[3] Framstad N C, O ksendal B, Sulem A. A sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance. Jour Optim Theory Appl, 2004, 121(1): 77--98 (Errata, 2005, 124(2): 511--512)

[4] James M R. Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games. Math Cont Sig Syst, 1992, 5: 401--417

[5] Lim A E B, Zhou X. A new risk-sensitive maximum principle. IEEE Trans Autom Cont, 2005, 50(7): 958--966

[6] Nagai H, Peng S. Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann Appl Proba, 2002, 12(1): 173--195

[7] Pham H. Optimal stopping of controlled jump diffusion processes: A viscosity solution approach. Jour Math Sys Esti Cont, 1998, 8(1): 1--27

[8] Shi J. Wu Z. Relationship between MP and DPP for the stochastic optimal control problem of jump diffusions. Appl Math Optim, DOI: 10.1007/S00245-010-9115-8, published on line Forward and backward stochastic optimal control theory with Poisson jumps and its applications. Shandong University PhD Thesis, 2009

[9] Situ R. A maximum principle for optimal controls of stochastic with random jumps//Proc National Conference on Control Theory and Its Applications. Qingdao, China, 1991

[10] Tang S, Li X. Necessary conditions for optimal control of stochastic systems with random jumps. SIAM Jour Cont Optim, 1994, 32(5): 1447--1475

[11] Whittle P. A risk-sensitive maximum principle. Sys Cont Lett, 1990, 15: 183--192

[12] Whittle P. A risk-sensitive maximum principle: The case of imperfect state observation. IEEE Trans Autom Cont, 1991, 36(7): 793--801

[13] Yong J, Zhou X. Stochastic Controls: Hamiltonian Systems and HJB Equations. New York: Springer-Verlag, 1999

[14] Zhou X. Sufficient conditions of optimality for stochastic systems with controllable diffusions. IEEE Trans Autom Cont, 1996, 41(8): 1176--1179

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 12

Metrics

本文评价

推荐阅读 10

[1]	李从明, 吴志刚. RADIAL SYMMETRY FOR SYSTEMS OF FRACTIONAL LAPLACIAN[J]. 数学物理学报(英文版), 2018, 38(5): 1567-1582.
[2]	曹文涛, 黄飞敏, 李天虹, 于慧敏. GLOBAL ENTROPY SOLUTIONS TO AN INHOMOGENEOUS ISENTROPIC COMPRESSIBLE EULER SYSTEM[J]. 数学物理学报(英文版), 2016, 36(4): 1215-1224.
[3]	闫奇姝. PERIODIC OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR PARABOLIC EQUATIONS WITH IMPULSE CONTROL[J]. 数学物理学报(英文版), 2016, 36(3): 847-862.
[4]	张启侠, 孙启良. A MAXIMUM PRINCIPLE APPROACH TO STOCHASTIC H₂/H_∞ CONTROL WITH RANDOM JUMPS[J]. 数学物理学报(英文版), 2015, 35(2): 348-358.
[5]	杨婉蓉, 酒全森. GLOBAL REGULARITY FOR MODIFIED CRITICAL DISSIPATIVE QUASI-GEOSTROPHIC EQUATIONS[J]. 数学物理学报(英文版), 2014, 34(6): 1741-1748.
[6]	邱红兵. AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL[J]. 数学物理学报(英文版), 2013, 33(6): 1561-1570.
[7]	田大平, 李光汉, 吴传喜. THE SURFACE AREA PRESERVING MEAN CURVATURE FLOW IN QUASI-FUCHSIAN MANIFOLDS[J]. 数学物理学报(英文版), 2012, 32(6): 2191-2202.
[8]	胡二彦. LOWER INEQUALITIES OF HEAT SEMIGROUPS BY USING PARABOLIC MAXIMUM PRINCIPLE[J]. 数学物理学报(英文版), 2012, 32(4): 1349-1364.
[9]	邵志强; 洪家兴. THE EIGENVALUE PROBLEM FOR THE LAPLACIAN EQUATIONS[J]. 数学物理学报(英文版), 2007, 27(2): 329-337.
[10]	陈静; 徐学文. EXISTENCE OF GLOBAL SMOOTH SOLUTION FOR SCALAR CONSERVATION LAWS WITH DEGENERATE VISCOSITY IN 2-DIMENSIONAL SPACE[J]. 数学物理学报(英文版), 2007, 27(2): 430-436.
[11]	汪更生. OPTIMAL CONTROL PROBLEM FOR PARABOLIC VARIATIONAL INEQUALITIES[J]. 数学物理学报(英文版), 2001, 21(4): 509-525.
[12]	张平健. A MAXIMUM PRINCIPLE FOR DISTRIBUTED PARAMETER SYSTEMS WITH MIXED PHASE-CONTROL CONSTRAINTS ANDENDPOINT CONSTRAINTS[J]. 数学物理学报(英文版), 1997, 17(2): 151-158.