A Pension Fund Model in the Accumulation Phase: a Stochastic Control Approach

In this paper we propose and study a continuous time stochastic model of optimal al- location for a defined contribution pension fund in the accumulation phase. 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 naturally formu- lated as an optimal stochastic control problem with state constraints and is treated by the dynamic programming approach. We show that the value function of the problem is a continuous viscosity solution of the associated Hamilton-Jacobi-Bellman equation. In the special case when the bound- ary is absorbing we show that it is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation