A pension fund in the accumulation phase: a stochastic control approach

Abstract

In the paper [6] the authors propose and study a continuous time stochastic model of optimal allocation for a defined contribution pension fund with minimum guarantee. Their target is to maximize the expected utility from current wealth over an infinite horizon, whereas usually portfolio selection models for pension funds maximize the expected utility from final wealth over a finite horizon (the retirement time). In this model the dynamics of wealth takes directly into account the flows of contributions and benefits; moreover the level of wealth is constrained to stay above a solvency level. The fund manager can invest in a riskless asset and in a risky asset but borrowing and short selling are prohibited. The model is formulated as an optimal stochastic control problem with constraint and is treated by the dynamic programming approach, showing that the value function of the problem is a regular solution of the associated Hamilton-Jacobi-Bellman equation. Then they apply verification techniques to get the optimal allocation strategy in feedback form and to study its properties, giving finally a special example with explicit solution. Nevertheless the aim of the authors is to study the problem starting from the time in which the first retirements of contributors occur (when the contribution flow becomes constant and the state equation homogeneous on time), leaving out the accumulation phase and the optimization problem in the first period. In our paper we describe the model and the problem in the accumulation phase, when the state equation is time-dependent. We will show that the value function is continuous and that it solves the associated Hamilton-Jacobi-Bellman equation in a viscosity sense. In the special case when the boundary is absorbing (therefore the value function is explicitally computable on this boundary and so a Dirichlet type condition is available for the boundary differential problem), we will show that it is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.