2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210208We find the minimum probability of lifetime ruin of an investor who can invest in a market with a risky and a riskless asset and can purchase a deferred annuity. Although we let the admissible set of strategies of annuity purchasing process to be increasing adapted processes, we find that the individual will not buy a deferred life annuity unless she can cover all her consumption via the annuity and have enough wealth left over to sustain her until the end of the deferral period.Key Words: Life annuities, retirement, optimal investment, stochastic control, free boundary problemOptimization and ControlProbabilityRisk Managementtext