Annuitizing at a bounded, absolutely continuous rate to minimize the probability of lifetime ruin

We minimize the probability of lifetime ruin in a deterministic financial and insurance model, although the investor's time of death is random, with an age-dependent force of mortality. By contrast with the traditional anything-anytime annuitization model (that is, individuals can annuitize any fraction of their wealth at anytime), the individual only purchases life annuity income gradually, using a bounded, absolutely continuous rate. As in the anything-anytime annuitization case, we find that it is optimal for the individual not to purchase additional annuity income when her wealth is less than a specific linear function of her existing annuity income, which we call the buy boundary. Interestingly, we find the buy boundary in our model is identical to the one in the anything-anytime annuitization model. However, there is a separate threshold, which we call the safe level. (This threshold degenerates to the buy boundary in the anything-anytime annuitization model.) When wealth is greater than the safe level, the minimum probability of lifetime ruin is zero; when wealth lies between the buy boundary and the safe level, the individual's best choice is to purchase annuity income at the maximum allowable rate.