A Pension Fund Model with Surplus: an Infinite Dimensional Stochastic Control Approach

In the paper [13] 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 constraints 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. By the way usually pension funds provide for their members also a surplus term as benefit besides a minimimum guarantee term. Such a term could depend on a contract a priori defined by the fund and it is reasoneable to think this contract as a function depending on the past wealth of the fund in the last period: higher is the performance of the fund in the last period, higher is the surplus payed to the fund members. From a mathematical point of view this unavoidably leads to a delay system, making the problem considerably more difficult to treat. In our paper we add to the model of [13] a surplus term, preserving its remaining setting. In Section 2 we give a survey on the model described in [13] and discuss different possible types of surplus contracts which could be introduced in the model. In particular, by technical reasons, we choose for our model a surplus contract which guarantees the concavity of the value function of the optimization problem; actually this contract is, from a financial point of view, less meaningful than other ones, which we hope to study in future works. In Section 3, after having given a formal setting of the stochastic optimization problem which we want to investigate, we pass to investigate some qualitative results on the value function: we prove results about finiteness, time-dependence, increasiness and concavity. Section 4 is devoted to rewrite the problem in an infinite-dimensional setting: following the approach contained, for example, in [37] for the deterministic case and in [6], [10], [23] for the stochastic case, we use the tool of the infinite-dimensional approach to study the problem, which essentially consists in defining a new state variable carrying the present and the past (in the last period) of the old state variable and a new state equation. Anyway, differently from the cited references, here we have to use a more specific approach, because in our case the delay term is non-linear and this forces us to work in a suitable subspace of the Hilbert space where we impose the infinite-dimensional problem living. This new state equation, which is an infinite-dimensional SDE, is the formal counter-part of the one-dimensional