Let $\phi(t,z)\in C^{1,2}$ is a function taking values in $\mathbb{R}^D$ , $\rho_{i,j}$ and $\mu_i$ be vector-valued functions and consider the non-linear PDE $$ \partial_t\Phi(t,z) - \phi(0,z) - \sum_i \mu_i \partial_{z_i}\Phi(t,z) - \frac{\sum_{i,j} \rho_{i,j}\partial_{z_i}\Phi(t,z)\partial_{z_j}\Phi(t,z) }{2} = \frac{\sum_{i,j} \rho_{i,j}\partial_{z_i,z_j}^2\Phi(t,z) }{2} $$ where $\Phi(t,z)\triangleq \int_0^t \phi(s,z)\,ds$.
If $\mu_i$ and $\rho_{i,j}$ are given is it possible to solve the given PDE for $\Phi$? It not can we calculate or at-least estimate the $L^{\infty}$ norm of $\phi$?
Potential Extra Assumption: If need be I may be able to assume (but would prefer not to) that the term on the Right-hand-side is 0.