Optimal insurance in a continuous-time model

We seek the optimal dynamic consumption, investment, and insurance strategies for an individual who seeks to maximize her expected discounted utility of consumption and bequest over a fixed or random horizon, such as her random future lifetime. Thus, we incorporate an insurable loss and random horizon into the classical consumption and investment framework of Merton. We determine that if the premium is proportional to the expected payout, then the optimal per-claim insurance is deductible insurance; thus, we extend this result for static models to our dynamic setting. We compute the value function and optimal controls for many examples and contrast their qualitative properties, including the impact of the investor's horizon (or mortality) on the optimal controls and the interaction between the demand for insurance and the risky asset. We employ the Markov Chain approximation method of Kushner for those examples for which closed form solutions are not available.