above specializes to when $\delta_i=1$. Indeed, if $\delta_i = 1$, then $\rho_i = 0$ and $\gamma_i = 1$, and the functions $a_i$ and $b_i$ defined in reduce to $a_i \equiv 0$ and $b_i \equiv 1/\epsilon_i$. Thus becomes $f_i(t) = \epsilon_i^{-1}(T- t)+1$, and becomes .

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**Completing the proof:** We now complete the proof, using the form we found above for the optimal control of player $i$ in response to the other players’ choices. Namely, the best response of player $i$ is given by the controls in , where $f_i$ is defined as in , and we have seen that these formulas are valid for both cases $\delta_i=1$ and $\delta_i \neq 1$.

Note that if we assume that the other consumption functions are positive and continuous, then the optimal feedback consumption that we have found is also positive and continuous on $[0,T]$. Now, to conclude the proof, note that the original choice of $(\pi_i, c_i)_{i=1}^n$ is a strong equilibrium if and only if for each $i = 1, ..., n$, we have $$\pi^{i,*} = \pi_i \hspace{0,5cm} \text{ and } \hspace{0,5cm} c^{i,*}_t = c_i(t) \hspace{0,5cm} \forall t \in [0,T],$$ where $(\pi^{i,*},c^{i,*})$ were given in .

We first address the investment policy. 