Optimal Control for Stochastic Delay Systems Under Model Uncertainty: A Stochastic Differential Game Approach

In this paper, we study a robust recursive utility maximization problem for time-delayed stochastic differential equation with jumps. This problem can be written as a stochastic delayed differential game. We suggest a maximum principle of this problem and obtain necessary and sufficient condition of optimality. We apply the result to study a problem of consumption choice optimization under model uncertainty.

[1]  Anis Matoussi,et al.  Maximization of Recursive Utilities: A Dynamic Maximum Principle Approach , 2011, SIAM J. Financial Math..

[2]  Xun Yu Zhou.  Sufficient conditions of optimality for stochastic systems with controllable diffusions , 1996, IEEE Trans. Autom. Control..

[3]  Robert J. Elliott,et al.  A Stochastic Maximum Principle for a Markov Regime-Switching Jump-Diffusion Model and Its Application to Finance , 2012, SIAM J. Control. Optim..

[4]  Salvatore Federico,et al.  A stochastic control problem with delay arising in a pension fund model , 2011, Finance Stochastics.

[5]  Shi JingTao.  Maximum principle of recursive optimal control problem for forward-backward stochastic delayed system with Poisson jumps , 2012 .

[6]  U. Haussmann,et al.  A stochastic maximum principle for optimal control of diffusions , 1986 .

[7]  H. Kushner.  On the stochastic maximum principle: Fixed time of control , 1965 .

[8]  M. Schweizer,et al.  A Stochastic Control Approach to a Robust Utility Maximization Problem , 2007 .

[9]  Larry G. Epstein,et al.  Stochastic differential utility , 1992 .

[10]  J. Bismut.  Conjugate convex functions in optimal stochastic control , 1973 .

[11]  Sergei Savin,et al.  On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects , 2007, J. Optimization Theory and Applications.

[12]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[13]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[14]  H. Kushner.  Necessary conditions for continuous parameter stochastic optimization problems , 1972 .

[15]  Zhen Wu,et al.  Maximum principle for the stochastic optimal control problem with delay and application , 2010, Autom..

[16]  Bernt Øksendal,et al.  Portfolio optimization under model uncertainty and BSDE games , 2011 .

[17]  Bernt Øksendal,et al.  Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty , 2014, J. Optim. Theory Appl..

[18]  J. Bismut.  An Introductory Approach to Duality in Optimal Stochastic Control , 1978 .

[19]  Shige Peng,et al.  Anticipated backward stochastic differential equations , 2007, 0705.1822.

[20]  Boualem Djehiche,et al.  On the Stochastic Maximum Principle in Optimal Control of Degenerate Diffusions with Lipschitz Coefficients , 2007 .

[21]  Alain Bensoussan,et al.  Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions , 1983 .

[22]  James D. Hamilton.  A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle , 1989 .

[23]  Larry G. Epstein,et al.  Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework , 1989 .

[24]  Bernt Øksendal,et al.  Maximum Principles for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps , 2009, SIAM J. Control. Optim..

[25]  Catherine Donnelly,et al.  Quadratic Risk Minimization in a Regime-Switching Model with Portfolio Constraints , 2012, SIAM J. Control. Optim..

[26]  Zhen Wu,et al.  Maximum principle for forward-backward stochastic control system with random jumps and applications to finance , 2010, J. Syst. Sci. Complex..

[27]  S. Peng.  A general stochastic maximum principle for optimal control problems , 1990 .

[28]  A. Giovannini,et al.  Risk Aversion and Intertemporal Substitution in the Capital Asset Pricing Model , 1989 .

[29]  P. Weil.  Nonexpected Utility in Macroeconomics , 1990 .

[30]  Anatoli F. Ivanov,et al.  Optimal control of stochastic differential delay equations with application in economics , 2008 .

[31]  B. Øksendal,et al.  A maximum principle for optimal control of stochastic systems with delay, with applications to finance. , 2000 .

[32]  Jiang-Hua Lu,et al.  Progress in Mathematics , 2013 .

[33]  F. Ramsey,et al.  THE MATHEMATICAL THEORY OF SAVING , 1928 .

[34]  Anne Gundel,et al.  Robust utility maximization for complete and incomplete market models , 2005, Finance Stochastics.

[35]  D. Duffie,et al.  Continuous-time security pricing: A utility gradient approach , 1994 .

[36]  S. Peng,et al.  A dynamic maximum principle for the optimization of recursive utilities under constraints , 2001 .

[37]  Huyen Pham,et al.  Continuous-time stochastic control and optimization with financial applications / Huyen Pham , 2009 .

[38]  B. Øksendal,et al.  Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations , 2011, Advances in Applied Probability.

[39]  Catherine Donnelly.  Sufficient Stochastic Maximum Principle in a Regime-Switching Diffusion Model , 2011 .

[40]  Mark H. A. Davis.  Lectures on Stochastic Control and Nonlinear Filtering , 1985 .

[41]  B. Øksendal,et al.  Applied Stochastic Control of Jump Diffusions , 2004, Universitext.

[42]  Anis Matoussi,et al.  Robust utility maximization in a discontinuous filtration , 2012 .

[43]  Sanjoy K. Mitter,et al.  Nonlinear Filtering and Stochastic Control , 1983 .

[44]  Hai-ping Shi.  Backward stochastic differential equations in finance , 2010 .