  We consider an individual or household endowed with an initial capital and an
income, modeled as a deterministic process with a continuous drift rate. At
first, we model the discounting rate as the price of a zero-coupon bond at zero
under the assumption of a short rate evolving as an Ornstein-Uhlenbeck process.
Then, a geometric Brownian motion as the preference function and an
Ornstein-Uhlenbeck process as the short rate are taken into consideration. It
is assumed that the primal interest of the economic agent is to maximise the
cumulated value of (expected) discounted consumption from a given time up to a
finite deterministic time horizon $T\in\R_+$ or, in a stochastic setting,
infinite time horizon. We find an explicit expression for the value function
and for the optimal strategy in the first two cases. In the third case, we have
to apply the viscosity ansatz.
