relative investment ${\pi_{0}(t) = 1-\hat{\pi}(t)'\mathbf{1}}$, where $\pi_{i}(t)$ denotes the fraction of wealth allocated to asset $i$ at time $t$. It is to satisfy $\int_{0}^{T} \pi_{i}(t)^{2} dt < \infty$, ${\mathbb{P}}$-a.s.. Moreover, let $(c(t))_{t \in [0,T]}$ denote a non-negative, progressively measurable, real-valued stochastic consumption rate process with $\int_{0}^{T} c(t) dt < \infty$, ${\mathbb{P}}$-a.s., and $(y(t))_{t \in [0,T]}$ a non-negative, deterministic income-rate process with $\int_{0}^{T} y(t) dt < \infty$. Those technical conditions are assumed to get a solution for the subsequently formulated stochastic problem. The dynamics of the investor’s wealth process $V = \left(V(t)\right)_{t \in [0,T]}$ under the strategy $(\pi,c)$ to initial wealth $V(0) = v_{0} > 0$, including liquid assets, consumption and income, is then given by $$\begin{aligned}
 \label{eq:SDE:V:y}
dV(t) = {} & V(t)  \left[\left(r + \hat{\pi}(t)' \left(\mu - r \mathbf{1}\right)\right) dt + \hat{\pi}(t)'\sigma dW(t)\right] - c(t) dt + y(t) dt.\end{aligned}$$ The relative investment in the risk-free asset is ${\pi_{0}(t) = 1 - \hat{\pi}(t)' \mathbf{1}}$. We consider the objective of maximizing expected utility of future terminal wealth and consumption, starting at time $0$ and ending at $T$. Hence the objective function to be maximized is $$\begin{aligned}
 \label{eq:ObjectiveFunction}
J(\pi,c;v_{0}) = {\mathbb{E}}\left[\int_{0}^{T} U_{1}(t,c(t)) dt + U_{2}(V(T))\right],\end{aligned}$$ where $v_{0} > 0$ denotes the initial endowment 