Optimal dividend problems with a risk probability criterion

This paper considers a discrete time optimal dividend payout problem with a risk probability criterion. Different from the expected discounted dividends that are widely studied in the existing literature, our emphasis is the risk probability that an insurance company's total discounted dividend fails to reach a given dividend goal by the time of ruin. We aim to find an optimal dividend policy to minimize such risk probability. More precisely, we establish a general model for optimal dividend problems based on a Markov decision process with varying discount factors and a probability criterion. The associated optimality equation and the existence of optimal policies under the probability criterion are investigated. For a special case with independently identically distributed incomes, we further characterize the properties of optimal value functions and the structures of optimal dividend policies. A value iteration‐type algorithm for computing value functions and optimal policies is developed. The convergence and error bound analyses of the algorithm are also derived. Finally, an optimal dividend policy in a concrete example is presented to demonstrate our main results.