Controllo Ottimo Stocastico Applicato ai Fondi Pensione

This thesis in Quantitative Finance and Insurance concerns the use of mathematical tools into pension schemes. In particular, it studies the Stochastic Optimal Control Theory and its potential applications during the accumulation phase of a fund in order to guarantee the desired annuity at retirement. The theoretical framework comprises the so-called Dynamic Programming Principle. Such principle yields to the Hamilton-Jacobi-Bellman Equation which, combined with the Verification Theorem, provides a way to obtain, under proper conditions of regularity, close-form solutions to the stochastic control process to be used to control the dynamics of the pension fund in order to fulfill the policyholder objectives. Using these techniques, the stochastic problem translates into a deterministic one, involving a partial differential equation. An ansatz is then used in order to transform it into a system of ordinary differential equations. The form of the guess function is usually the hardest step of any stochastic optimal control. Therefore, finding a closed-form solution is not always possible. In this thesis, we have defined a stochastic control problem related to a pension fund in the accumulation phase, and we have been able to perform the arduous task of deriving its analytical solution. Once such a closed-form solution was obtained, a simulation analysis was performed in order to better grasp the dynamics embedded into the accumulation phase, related to the stochastic performance of financial markets in which the fund is invested.