# H–J–B Equations of Optimal Consumption-Investment and Verification Theorems

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## Abstract

We consider a consumption-investment problem on infinite time horizon maximizing discounted expected HARA utility for a general incomplete market model. Based on dynamic programming approach we derive the relevant H–J–B equation and study the existence and uniqueness of the solution to the nonlinear partial differential equation. By using the smooth solution we construct the optimal consumption rate and portfolio strategy and then prove the verification theorems under certain general settings.

## Keywords

Utility maximization Risk-sensitive stochastic control Factor models H–J–B equation Infinite time horizon## Mathematics Subject Classification

35J60 49L20 60F10 91B28 93E20## Notes

### Acknowledgments

The author would like to express his thanks to professor S. J. Sheu who attracted his interest to the consumption-investment problems with useful discussions about them, and also encouraged him to extend his private notes on the problems to an article to be published somewhere. Thanks are also due to the referee for his (her) careful reading of the manuscript and correcting a number of typos.

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