of the investor. All expectations in this paper are with respect to the real-world measure ${\mathbb{P}}$. The general portfolio optimization problem with initial wealth ${V(0) = v_{0} > 0}$ to be solved is then given by $$\begin{aligned}
 \label{eq:OptimizationProblem}
\mathcal{V}(v_{0}) = \sup_{(\pi,c) \in \Lambda} J(\pi,c;v_{0})\end{aligned}$$ subject to . $\mathcal{V}(v_{0})$ is the value function of the problem. $\Lambda$ denotes the set of admissible strategies $(\pi,c)$ such that ${V(t) + \int_{t}^{T} e^{- r (s-t)} y(s) ds \ge 0}$, ${\mathbb{P}}$-a.s., $\forall t \in [0,T]$, and which admit a unique solution to while satisfying the integrability condition ${{\mathbb{E}}\left[\int_{0}^{T} |U_{1}(t,c(t))| dt\right] < \infty}$. The so-called budget constraint reads $$\begin{aligned}
 \label{eq:BudgetConstraint:y}
{\mathbb{E}}\left[\int_{0}^{T} \tilde{Z}(t) c(t) dt + \tilde{Z}(T) V(T)\right] \le v_{0} + {\mathbb{E}}\left[\int_{0}^{T} \tilde{Z}(t) y(t) dt\right] = v_{0} + \int_{0}^{T} e^{-r t} y(t) dt.\end{aligned}$$ It describes the requirement that today’s value of future consumption and terminal wealth, less income, must not exceed the initial endowment. It can be shown that for the optimal $(\hat{\pi}^{\star}, c^{\star})$ to Problem , Equation holds with equality. We consider a preference utility model given by the utility functions $$\begin{aligned}
\begin{split} \label{eq:utilitymodel:new}
U_{1}(t,c) = {} & \left(e^{- \beta  t} a(t)\right)  \frac{1-b(t)}{b(t)}  \left(\frac{1}{1-b(t)}  \left(c - \bar{c}(t)\right)\right)^{b(t)}, \\
U_{2}(v) = {} & e^{- \beta  T} \hat{a}  