##
**Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon.**
*(English)*
Zbl 1042.91048

The authors consider an optimal investment problem for a factor model as a risk-sensitive stochastic control problem, where the mean returns of individual securities are explicitly affected by economic factors defined as Gaussian processes. The market considered has \(m+1\geq 2\) securities and \(n\geq 1\) factors. It is assumed that the set of securities includes one bond whose price is defined by the ordinary differential equation
\[
dS^0(t)=r(t)S^0(t) \,dt
\]
with a deterministic function \(r(t)\), and the other security prices and factors are assumed to satisfy stochastic differential equations
\[
dS^i(t)=S^i(t) \left[(a+AX_t)^i dt + \sum_{k=1}^{n+m}\sigma_k^i dW_t^k\right],
\]
and
\[
dX_t=(b+BX_t)dt+\Lambda dW_t.
\]
The investment strategies are supposed to be chosen without using information on factor processes, but by using only past information on security prices. The problem is formulated as a kind of stochastic control problem with partial information. The results concerning a finite-time horizon are summarized. Then the asymptotic behavior of the solution \(U(t,T)\) of the related inhomogeneous Riccati differential equation with the terminal condition is studied. Specific feature of asymptotics of the solution \(U(t,T)\) is stability as \(t\) and \(T-t\) tend to infinity, from which the asymptotic behavior of the value function and the optimal strategy of the problem on an infinite time horizon are obtained.

Reviewer: Yuliya S. Mishura (Kyïv)

### MSC:

91G10 | Portfolio theory |

93E20 | Optimal stochastic control |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

### Keywords:

Ito calculus; partial differential equation; optimal investment problem; factor model; risk-sensitive stochastic control problem
PDF
BibTeX
XML
Cite

\textit{H. Nagai} and \textit{S. Peng}, Ann. Appl. Probab. 12, No. 1, 173--195 (2002; Zbl 1042.91048)

Full Text:
DOI

### References:

[1] | BENSOUSSAN, A.(1992). Stochastic Control of Partially Observable Systems. Cambridge Univ. Press. · Zbl 0776.93094 |

[2] | BENSOUSSAN, A. and VAN SCHUPPEN, J. H. (1985). Optimal control of partially observable stochastic systems with an exponential-of integral performance index. SIAM J. Control Optim. 23 599-613. · Zbl 0574.93067 |

[3] | BIELECKI, T. R. and PLISKA, S. R. (1999). Risk-sensitive dynamic asset management. Appl. Math. Optim. 39 337-360. · Zbl 0984.91047 |

[4] | BIELECKI, T. R. and PLISKA, S. R. (2000). Risk-sensitive intertemporal CAPM, with application to fixed income management. · Zbl 0982.91024 |

[5] | BUCY, R. S. and JOSEPH, P. D. (1987). Filtering for Stochastic Processes with Applications to Guidance. Chelsea, New York. · Zbl 0174.21903 |

[6] | EL KAROUI, N. and QUENEZ, M.-C. (1995). Dynamic Programming pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29-66. · Zbl 0831.90010 |

[7] | FLEMING, W. H. (1995). Optimal Investment models and risk-sensitive stochastic control. IMA Vol. Math. Appl. 65 75-88. · Zbl 0841.90016 |

[8] | FLEMING, W. H. and RISHEL, R. (1975). Optimal Deterministic and Stochastic Control. Springer, Berlin. · Zbl 0323.49001 |

[9] | FLEMING, W. H. and SHEU, S. J. (1999). Optimal long term growth rate of expected utility of wealth. Ann. Appl. Probab. 9 871-903. · Zbl 0962.91036 |

[10] | FLEMING, W. H. and SHEU, S. J. (2000). Risk-sensitive control and an optimal investment model. Math. Finance 10 197-213. · Zbl 1039.93069 |

[11] | FLEMING, W. H. and SHEU, S. J. (2000). Risk-sensitive control and an optimal investment model (II). · Zbl 1039.93069 |

[12] | KUCERA, V. (1972). A contribution to matrix quadratic equations. IEEE Trans. Automat. Control AC-17 344-347. · Zbl 0262.93043 |

[13] | KURODA, K. and NAGAI, H. (2000). Risk-sensitive portfolio optimization on infinite time horizon. Stochastics Stochastics Rep. · Zbl 1041.91036 |

[14] | NAGAI, H. (2000). Risk-sensitive dynamic asset management with partial information. In Stochastics in Finite and Infinite Dimension. A Volume in Honor of Gopinath Kallianpur (T. Hida, R. L. Karandikar, H. Kunita, B. S. Rajput, S. Watanabe and J. Xiong, eds.) 321-340. Birkhäuser, Boston. · Zbl 1013.91057 |

[15] | WONHAM, W. M. (1968). On a matrix Riccati equation of stochastic control. SIAM J. Control Optim. 6 681-697. · Zbl 0182.20803 |

[16] | TOYONAKA, 560-8531 JAPAN E-MAIL: nagai@sigmath.es.osaka-u.ac.jp DEPARTMENT OF MATHEMATICS SHANGDONG UNIVERSITY JINAN 250100 CHINA E-MAIL: peng@sdu.edu.cn |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.