Optimal annuitization under piecewise deterministic mortality force

Purchasing an annuity is one of the most significant financial choices individuals can make for retirement. This decision marks the point where retirees convert their savings into a steady, guaranteed income stream for life, prioritizing long-term financial stability over the potential growth of other investments. The uncertainty surrounding lifespan complicates this choice, as individuals must consider how to ensure their resources last throughout their lifetime. This Thesis explores the optimal timing for an individual to purchase an annuity while facing two sources of uncertainty: fluctuations in financial markets and the random evolution of his survival probability. We consider an individual whose retirement wealth is invested in a financial fund and who may choose a random time to irreversibly convert the entire wealth into an immediate lifetime annuity. The individual's mortality force evolves according to a piecewise deterministic Markov process, where the mortality risk remains constant between random jump times, with the size of each jump governed by a probability distribution. Coupled with financial market fluctuations, this introduces a complex decision-making environment where the retiree must determine the best time to convert his wealth into an annuity. The problem is initially formulated as a three-dimensional continuous-time optimal stopping problem. However, by exploiting the particular structure of the piecewise deterministic Markov process, we are able to reduce the complexity. Specifically, we transform it into a sequence of nested one-dimensional optimal stopping problems. This transformation simplifies the mathematical analysis and enables us to derive explicit solutions. The solution to the problem hinges on analyzing two key regions: the continuation region, where the individual chooses to keep his wealth invested in the financial fund, and the stopping region, where he opts to purchase the annuity. The boundary between these regions is critical in determining the optimal timing of the annuity purchase. Initially, we focus on a simplified scenario where the force of mortality remains constant throughout the individual's life. This case allows us to derive explicit solutions and serves as a foundation to prove properties that will be generalized later for the more general model. In the general scenario, where the mortality force is modelled as a piecewise deterministic Markov process, we examine the properties of the value function and analyze the geometry of the continuation and the stopping regions. The shape of these regions varies depending on the presence of incentives or taxes on annuity purchases. Then, we investigate a special case in which the mortality force experiences a single jump at a random time. This scenario yields explicit solutions that provide insights into the financial consequences of sudden changes in mortality risk.