subtracted as it reduces the time $t$ value of the minimum required capital.

Solution: Separation technique {#sec:SeparationTechnique}
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In the sequel we follow the separation technique approach by [@KaratzasShreve1998] and [@LaknerNygren2006] for solving the consumption-terminal wealth problem as defined by . We split the problem into two sub-problems: the consumption-only and terminal wealth-only problem. Both individual problems are separately solved via the martingale method, similar to the approach by [@Aase2017]. The individual problem solutions are optimally merged at the end. For this sake, let us consider the two individual problems first.

The consumption problem {#sec:ConsumptionProblem:y}
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The consumption-only problem is $$\begin{aligned}
\begin{split} \label{eq:OptimizationProblem:ConsumptionOnly}
J_{1}(\pi,c;v_{1}) = {} & {\mathbb{E}}\left[\int_{0}^{T} U_{1}(t,c(t)) dt\right], \\
\mathcal{V}_{1}(v_{1}) = {} & \sup_{(\pi,c) \in \Lambda_{1}} J_{1}(\pi,c;v_{1})
\end{split}\end{aligned}$$ subject to the budget constraint $$\begin{aligned}
 \label{eq:BudgetConstraint:ConsumptionOnly:y}
{\mathbb{E}}\left[\int_{0}^{T} \tilde{Z}(t) c(t) dt\right] \le v_{1} + {\mathbb{E}}\left[\int_{0}^{T} \tilde{Z}(t) y(t) dt\right] = v_{1} + \int_{0}^{T} e^{-r t} y(t) dt.\end{aligned}$$ $\Lambda_{1}$ 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 ${{\mathbb{E}}\left[\int_{0}^{T} |U_{1}(t,c(t))| dt\right] < \infty}$.

[@Steffensen2011] provides a proof for CRRA utility functions by solving the associated Hamilton-Jacobi-Bellman (HJB) equation. 