A stochastic control problem with delay arising in a pension fund model

This paper deals with the optimal control of a stochastic delay differential equation arising in the management of a pension fund with surplus. The problem is approached by the tool of a representation in infinite dimension. We show the equivalence between the one-dimensional delay problem and the associated infinite-dimensional problem without delay. Then we prove that the value function is continuous in this infinite-dimensional setting. These results represent a starting point for the investigation of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation in the viscosity sense and for approaching the problem by numerical algorithms. Also an example with complete solution of a simpler but similar problem is provided.