Suppose I have a PDE of the form $$ 0=-\partial_t f(t,z) + \phi(0,z)+ \sum_{i=1}^{\infty} b_i(t,z)\partial_{z_i}f(t,z) + \frac{1}{2}\sum_{i,j=1}^{\infty} a_{i,j}(t,z)(\partial_{z_j}\partial_{z_i}f(t,z) - g(\partial_{z_i}\partial_{z_j}f(t,z)), $$ where $g$ is a $C^{\infty}$, strictly convex function on $\mathbb{R}$.
What conditions do I need to impose on $a_{i,j}$, $b_i$ and $g$ in order to assure that the above PDE has a (preferably unique) mild viscosity solution?
Note: (If this problem is too difficult, we may assume that the $a_{i,j}$, $b_i$ and $g$ are continuous, but I prefer not to have to).