We investigate a class of optimal consumption problems where the wealth X(t) at time t is given by a stochastic differential delay equation with a parameter. Not only the present value X(t), but also X(t-\delta) and some sliding average of previous values affects the growth at time t. Two cases are considered: 1) The parameter is a given deterministic function, giving a stochastic control problem with complete observations. 2) The parameter is an unobserved random variable, giving a stochastic control problem with partial observations. In this case, filtering theory is used to reduce the problem to a completely observed one.
In both cases, due to the delay, the resulting dynamic programming problems are in general infinite dimensional. Because of the specific structure of the dependence of the past that we consider, we are able to reduce the problem to finite dimensions. A verification theorem of variational inequality type is proved and applied to solve explicitly the control problems. (Explicit formulas for the value functions and the optimal consumption rates are given.)