for $\lambda_{1}$ as follows.

\[thm:Solution:ConsumptionOnly:ValueFunction:lambda\] The optimal value function $\mathcal{V}_{1}(v_{1})$ to Problem is strictly increasing and concave in $v_{1}$. Its value and first and second derivative with respect to the initial budget $v_{1}$ are given by $$\begin{aligned}
\mathcal{V}_{1}(v_{1}) = {} & \int_{0}^{T} \frac{1-b(t)}{b(t)} \left(\frac{e^{\left[\beta - b(t) \left(r - \frac{1}{2} \frac{1}{b(t)-1} \|\gamma\|^{2}\right)\right] t}}{a(t)}\right)^{\frac{1}{b(t)-1}} \lambda_{1}^{\frac{b(t)}{b(t)-1}} dt, \\
\mathcal{V}_{1}^{\prime}(v_{1}) = {} & \lambda_{1} > 0, \\
\mathcal{V}_{1}^{\prime\prime}(v_{1}) = {} & \lambda_{1}^{\prime} = - \left(\int_{0}^{T} \left(\frac{e^{\left[\beta - b(t) \left(r - \frac{1}{2} \frac{1}{b(t)-1} \|\gamma\|^{2}\right)\right] t}}{a(t)}\right)^{\frac{1}{b(t)-1}} \lambda_{1}^{- \frac{b(t)-2}{b(t)-1}}  dt\right)^{-1} < 0.\end{aligned}$$

The terminal wealth problem {#sec:TerminalWealthProblem}
---------------------------

The terminal wealth-only problem is $$\begin{aligned}
\begin{split} \label{eq:OptimizationProblem:TerminalWealthOnly}
J_{2}(\pi,c;v_{2}) = {} & {\mathbb{E}}\left[U_{2}(V(T))\right], \\
\mathcal{V}_{2}(v_{2}) = {} & \sup_{(\pi,c) \in \Lambda_{2}} J_{2}(\pi,c;v_{2})
\end{split}\end{aligned}$$ subject to the budget constraint $$\begin{aligned}
 \label{eq:BudgetConstraint:TerminalWealthOnly}
{\mathbb{E}}\left[\tilde{Z}(T) V(T)\right] \le v_{2},\ v_{2} \ge 0.\end{aligned}$$ $\Lambda_{2}$ denotes the set of admissible strategies $(\pi,c)$ such that ${V(t) \ge 0}$, ${\mathbb{P}}$-a.s., $\forall t \in [0,T]$, and which admit a unique solution to for ${y(t) \equiv 0}$.

In order to guarantee the terminal wealth floor, note $V(T) > F$, let us assume the following lower bound for $v_{2}$ which equals the discounted terminal floor: $$\begin{aligned}
v_{2} > {} & e^{- r T} F =: F_{2}(0),\ F_{2}(t) := {\mathbb{E}}\left[\frac{\tilde{Z}(T)}{\tilde{Z}(t)} F \Big| 