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**Optimal consumption and investment problem with random horizon in a BMAP model.**
*(English)*
Zbl 1314.91192

Summary: In this paper, we consider the consumption and investment problem with random horizon in a Batch Markov Arrival Process (BMAP) model. The investor invests her wealth in a financial market consisting of a risk-free asset and a risky asset. The price processes of the riskless asset and the risky asset are modulated by a continuous-time Markov chain, which is the phase process of a BMAP. The possible consumption or investment are restricted to a sequence of random discrete time points which are determined by the same BMAP. The investor has only consumption opportunities at some of these random time points, has both consumption and investment opportunities at some other random time points, and can do nothing at the remaining random time points. The object of the investor is to select the consumption-investment strategy that maximizes the expected total discounted utility. The purpose of this paper is to analyze the impact of the consumption-investment opportunity and the economic state on the value functions and consumption-investment strategies. The general solution and the exact solution under the assumption that the consumption and the terminal wealth are evaluated by the power utility are obtained. Finally, a numerical example is presented.

### MSC:

91G10 | Portfolio theory |

60K20 | Applications of Markov renewal processes (reliability, queueing networks, etc.) |

90C40 | Markov and semi-Markov decision processes |

### Keywords:

optimal consumption and investment; random horizon; BMAP; Bellman equation; Markov decision process
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\textit{X. Chen} and \textit{X.-q. Yang}, Insur. Math. Econ. 61, 197--205 (2015; Zbl 1314.91192)

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