## Abstract

We study the stochastic control problem of maximizing expected utility from terminal wealth, when the wealth process is subject to shocks produced by a general marked point process; the problem of the agent is to derive the optimal allocation of his wealth between investments in a nonrisky asset and in a (costly) insurance strategy which allows "lowering" the level of the shocks. The agent's optimization problem is related to a suitable dual stochastic control problem in which the constraint on the insurance strategy disappears. We establish a general existence result for the dual problem as well as the duality between both problems. We conclude by some applications in the context of power (and logarithmic) utility functions and linear insurance premium which show, in particular, the existence of two critical values for the insurance premium: below the lower critical value, agents prefer to be completely insured, whereas above the upper critical value they take no insurance.

Original language | English (US) |
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Pages (from-to) | 283-312 |

Number of pages | 30 |

Journal | Annals of Applied Probability |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2000 |

## Keywords

- Convex analysis
- Duality
- Optimal insurance
- Optional decomposition
- Stochastic control

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty