An optimal consumption problem in finite time with a constraint on the ruin probability

In this paper, we investigate the following problem: For a given upper bound for the ruin probability, maximize the expected discounted consumption of an investor in finite time. The endowment of the agent is modeled by a Brownian motion with positive drift. We give an iterative algorithm for the solution of the problem, where in each step an unconstrained, but penalized problem is solved. For the discontinuous value function of the penalized problem, we show that it is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. Moreover, we characterize the optimal strategy as a barrier strategy with continuous barrier function.