\begin{align}
\tilde{v}(r_t, t) 
&= - \partial_t \log \tilde{\kappa}(t) \log_{r_t}(\hat{r}_1) 
= - \frac{- \partial_t \log \tilde{\kappa}(t)}{-\partial_t \log \kappa(t)} \partial_t \log \kappa(t) \log_{r_t}(\hat{r}_1) \\
&= - c (1 - t) \partial_t \log \kappa(t) \log_{r_t}(\hat{r}_1) 
= c(1-t) ~v(r_t, t)
= \lambda(t) ~v(r_t, t).
\end{align}
