## 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.

### MSC:

 91G10 Portfolio theory 93E20 Optimal stochastic control 60H30 Applications of stochastic analysis (to PDEs, etc.)
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### 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
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