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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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[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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