While deriving the SHJB equation for a specific case I stumbled upon this (using Einstein's convention for repeated indices):
$$\rho J = (x_i \tilde u_i-K_i) + (\alpha_i + \beta_{ij}x_j - R_{ijk} \tilde u_j \tilde u_k)\partial_{x_i}J+ \frac{1}{2}\frac{\partial^2 J}{\partial x_i \partial x_j} \nu_{ij}$$ $\alpha, \beta$ and $\nu$ are constant matrices and $\tilde u _i$ satisfies $R_{ijk}\partial_{x_i} J \tilde u_j = \frac{x_k}{2}$. Does somebody have any references on this? The matrix $(R_i)_{jk}$ is positive definite symmetric. If you substitute you get that the non linear part satisfies $$p(\nabla J)=\frac{1}{4}x_i(S^{-1})_{ij}x_j$$ Where $S^{-1}$ is the inverse of $\partial _{x_i}JR_{ijk}$.
