State-dependent correlation investment problem ($\rho =\rho (M)$)

When asset correlation varies with a traded market index, the index is both an asset and a state variable. Writing $m=\log M$ and optimizing Brownian portfolio exposures accounts for the cross derivative between wealth and $m$. Power homogeneity reduces the Hamilton-Jacobi-Bellman equation to an ordinary differential equation in $m$.

For a complete-market investment-consumption problem with constant relative risk aversion utility, the nonlinear wealth-homogeneity coefficient equation may contain $f^{\prime 2}/f$. Writing $f=g^R$ cancels this gradient square against the one from $f^{\prime \prime }$. In the index-driven correlation model the result is ${\sigma }_0^2{g}^{\prime \prime }/2+B{g}^{\prime }-\delta (m)g+1=0$, a linear differential equation; the positive economic solution gives consumption $w/g$.