\begin{equation}\label{optimalstrategies}
\pi^{*}_t  = \frac{\lambda_{t}+\sigma_{t}(\rho_{t} Z_{t}+\hat{\rho}_{t} \hat{Z}_{t})}{\gamma\sigma_{t}^{2}}\quad\hbox{and}\quad \frac{c^*_t}{X^*_t}=\tilde{c}^{*}_t = \delta^{\psi}e^{-\frac{\psi}{\theta}D_{t}}\qquad \forall t\in[0,\tau),
\end{equation}
