v_{1}) \tilde{Z}(t)^{\frac{1}{b(t)-1}} + \bar{c}(t) = (1-b(t)) \left(\lambda_{1} \frac{e^{\beta  t}}{a(t)} \tilde{Z}(t)\right)^{\frac{1}{b(t)-1}} + \bar{c}(t), \\
V_{1}(t ; v_{1}) = {} & \int_{t}^{T} g(s,t; v_{1}) \tilde{Z}(t)^{\frac{1}{b(s)-1}} ds + F_{1}(t), \\
V_{1}(T ; v_{1}) = {} & 0,\end{aligned}$$ for all $t \in [0,T]$, where $$\begin{aligned}
g(s,t; v_{1}) = (1-b(s)) \left(\frac{e^{\beta  s - b(s) \left(r - \frac{1}{2} \frac{1}{b(s)-1} \|\gamma\|^{2}\right) (s-t)}}{a(s)}\right)^{\frac{1}{b(s)-1}} \lambda_{1}^{\frac{1}{b(s)-1}}.\end{aligned}$$ $\lambda_{1} = \lambda_{1}(v_{1}) > 0$ satisfies the budget constraint uniquely and is subject to the equation $$\begin{aligned}
 \label{eq:ConsumptionOnly:lambda}
\int_{0}^{T} g(t,0; v_{1}) dt = v_{1} - F_{1}(0).\end{aligned}$$ $\tilde{t}_{1} = \tilde{t}_{1}(v_{1}) \in (t,T)$ is the solution to the equation $$\begin{aligned}
 \label{eq:ConsumptionOnly:tautilde}
\int_{t}^{T} \frac{1}{b(s)-1} g(s,t; v_{1}) \tilde{Z}(t)^{\frac{1}{b(s)-1}} ds = \frac{1}{b(\tilde{t}_{1})-1} \int_{t}^{T} g(s,t; v_{1}) \tilde{Z}(t)^{\frac{1}{b(s)-1}} ds.\end{aligned}$$ For the optimal $c_{1}(t;v_{1})$, Equation is fulfilled with equality.

We remind the reader that all proofs can be found in \[app:Proofs\]. It is clear that ${c_{1}(t;v_{1}) > \bar{c}(t)}$, a.s.. We now aim to interpret the optimal investment strategy as proportional portfolio insurance (PPI) strategy. The first strategy family corresponds to a constant multiple, the latter one is more general and also covers proportional strategies with time-varying or even state-dependent multiples. [@ZielingMahayniBalder2014] evaluate the performance of such strategies. Theorem \[thm:Solution:ConsumptionOnly\] shows that the optimal investment strategy generally is a PPI strategy with time-varying 