Annuitization and asset allocation under exponential utility

Because maximizing exponential utility generally does not prevent wealth from dropping below 0, we restrict the investment, consumption, and annuitization strategies so that wealth remains non-negative. We solve the optimization problem via stochastic control and obtain semi-explicit solutions by using the Legendre dual. We prove that the optimal annuitization strategy is a barrier strategy. We also provide some numerical examples to illustrate our results and to analyze their sensitivity to the parameters.