Dynamic asset management with risk-sensitive criterion and non-negative factor constraints: a differential game approach

In this paper, we study continuous time portfolio optimization problem where individual securities are directly affected by economic factors. We consider the risk-sensitive criterion function as is familiar in the robust control literature. This is the natural setting for studying the infinite horizon case of the control problem arising in portfolio optimization. Our result extends earlier works by imposing explicitly the non-negativity constraint on the economic factors. This is achieved by using reflected diffusions. The risk-sensitive control problem with reflected diffusion is then converted into a stochastic differential game. The lower value of this game leads immediately to the desired optimal strategy. Also we prove the existence of unique strong solution to reflected diffusions with bounded measurable drift coefficient which is the first result of its kind for higher dimensional reflected diffusions.

[1]  P. Dupuis,et al.  SDEs with Oblique Reflection on Nonsmooth Domains , 2008 .

[2]  M. K. Ghosh,et al.  A nonzero-sum stochastic differential game in the orthant , 2005 .

[3]  Tomasz R. Bielecki,et al.  Economic Properties of the Risk Sensitive Criterion for Portfolio Management , 2003 .

[4]  M. K. Ghosh,et al.  A Stochastic Differential Game in the Orthrant , 2002 .

[5]  S. Pliska,et al.  Risk-Sensitive Dynamic Asset Management , 1999 .

[6]  G. Gustafson,et al.  Boundary Value Problems of Mathematical Physics , 1998 .

[7]  H. Nagai.  Bellman equations of risk sensitive control , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[8]  M. Shashiashvili.  A lemma of variational distance between maximal functions with application to the skorokhod problem in a nonnegative orthant with state-dependent reflection directions , 1994 .

[9]  M. James,et al.  Risk-sensitive control and dynamic games for partially observed discrete-time nonlinear systems , 1994, IEEE Trans. Autom. Control..

[10]  C. Costantini,et al.  The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations , 1992 .

[11]  P. Dupuis,et al.  On oblique derivative problems for fully nonlinear second-order elliptic PDE’s on domains with corners , 1991 .

[12]  Vivek S. Borkar,et al.  Optimal Control of Diffusion Processes , 1989 .

[13]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[14]  A. Veretennikov.  On Strong and Weak Solutions of One-Dimensional Stochastic Equations with Boundary Conditions , 1982 .

[15]  A. Bensoussan.  Stochastic control by functional analysis methods , 1982 .

[16]  R. C. Merton,et al.  Optimum Consumption and Portfolio Rules in a Continuous-Time Model* , 1975 .

[17]  V. Benes.  Existence of Optimal Strategies Based on Specified Information, for a Class of Stochastic Decision Problems , 1970 .

[18]  I. N. Sneddon,et al.  Boundary value problems , 2007 .

[19]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[20]  K. Fan.  Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces. , 1952, Proceedings of the National Academy of Sciences of the United States of America.