Power-wealth investment with running utility (${\pi }^{\ast }/W=(\mu -r)/((1-\gamma ){\sigma }^2)$)

For a self-financing portfolio earning both running reward $e^{-\rho t}{W}^{\gamma }$ and terminal reward $W_T^{\gamma }$, $0<\gamma <1$, with no consumption withdrawals, Hamilton-Jacobi-Bellman equation gives $J(W,t)=A(t)W^{\gamma }$ and the displayed constant risky fraction. The coefficient satisfies $A^{\prime }+\kappa A+e^{-\rho t}=0$, $A(T)=1$, where $\kappa =\gamma r+\gamma (\mu -r{)}^2/(2(1-\gamma ){\sigma }^2)$. Its positive solution is $e^{\kappa (T-t)}+{\int }_t^T{e}^{\kappa (s-t)-\rho s}ds$. Risk aversion is $1-\gamma$; increasing $\gamma$ raises risky investment and can require borrowing.