floor $F_{1}(t)$ at time $t$, equal to the time $t$ value of the accumulated outstanding future consumption floor minus income. Notice that $\tilde{t}_{1}$ can firstly be determined at time $t$, since the value depends on the stochastic $\tilde{Z}(t)$ which is not known before time $t$. Hence, $\tilde{t}_{1}$ is time- and also state-dependent and thus the optimal PPI strategy itself is time- and state-dependent through its PPI multiple. The PPI multiple in summary is time-varying, state-dependent and depends on all future coefficients of risk aversion via $b(\tilde{t}_{1})$.

Furthermore, ${V_{1}(0 ; v_{1}) > F_{1}(0)}$ holds by the assumption in . In addition, $\hat{\pi}_{1}(t ; v_{1})$ converges to $0$ when ${V_{1}(t ; v_{1})}$ approaches $F_{1}(t)$. Thus, ${V_{1}(t ; v_{1}) > F_{1}(t)}$ a.s., which additionally follows directly from the formula for $V_{1}(t ; v_{1})$ in Theorem \[thm:Solution:ConsumptionOnly\]. This further implies that $(\hat{\pi}_{1},c_{1})$ is an admissible pair, i.e. ${(\hat{\pi}_{1},c_{1}) \in \Lambda_{1}}$. The next remark provides the solution under time-independent risk aversion.

When $b(t) \equiv b$, then $$\begin{aligned}
\hat{\pi}_{1}(t ; v_{1}) = \frac{1}{1 - b} \Sigma^{-1} (\mu - r \mathbf{1}) \frac{V_{1}(t ; v_{1}) - F_{1}(t)}{V_{1}(t ; v_{1})}\end{aligned}$$ which is a conventional CPPI strategy with constant multiple. Moreover, if $\bar{c}(t) - y(t) \equiv 0$, i.e. the minimum 