Optimal consumption and portfolio policies when asset prices follow a diffusion process.

*(English)*Zbl 0678.90011Summary: We consider a consumption-portfolio problem in continuous time under uncertainty. A martingale technique is employed to characterize optimal consumption-portfolio policies when there exist nonnegativity constraints on consumption and on final wealth. We also provide a way to compute and verify optimal policies. Our verification theorem for optimal policies involves a linear partial differential equation, unlike the nonlinear partial differential equation of dynamic programming. The relationship between our approach and dynamic programming is discussed. We demonstrate our technique by explicitly computing optimal policies in a series of examples. In particular, we solve the optimal consumption-portfolio problem for hyperbolic absolute risk aversion utility functions when the asset prices follow a geometric Brownian motion. The optimal policies in this case are no longer linear when nonnegativity constraints on consumption and on final wealth are included. By these examples, one can see that our approach is much easier than the dynamic programming approach.

##### MSC:

91G10 | Portfolio theory |

49L20 | Dynamic programming in optimal control and differential games |

60J60 | Diffusion processes |

90C39 | Dynamic programming |

##### Keywords:

consumption-portfolio problem; continuous time; uncertainty; martingale technique; linear partial differential equation; hyperbolic absolute risk aversion utility functions; geometric Brownian motion; continuous-time
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\textit{J. C. Cox} and \textit{C.-f. Huang}, J. Econ. Theory 49, No. 1, 33--83 (1989; Zbl 0678.90011)

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