I've been looking into stochastic optimal control problems as part of my thesis and have been struggling to find an analytic solution to the semi-linear PDE in $d$ dimensions:
$ \partial_t \Phi(v,t) - \frac{1}{4} \beta(t)^2 (\nabla f(v)^T \nabla_v \Phi(v,t))^2 + \frac{1}{2} \eta \beta(t)^2 \text{tr}\{ \Sigma \nabla_v^2 \Phi(v,t) \} = 0 $
with boundary condition $\Phi(v,T) = f(v)$.
$\beta(t)$ is a known function, but could also for simplicity be set to a constant value, and $\eta$ and $\sigma$ are also pre-set constants. $f(v)$ is another known function, we could for instance assume it to be a simple quadratic $f(v) = \frac{1}{2} v^T A v$ with semi-positive definite matrix $A$.
It would be enough for me to know, if an analytic solution exists in the simplified one-dimensional case ($f(v) = \frac{1}{2} v^2$, $\beta(t) = 1$, $\eta = \sigma = 1$):
$ \partial_t \Phi(v,t) - \frac{1}{4} v^2 (\partial_v \Phi(v,t))^2 + \frac{1}{2} \partial_{vv} \Phi(v,t) = 0 $.
I checked the literature and the expression looks very similar to the case, where the Hopf-Cole transformations could be used. But in my case there is the $v^2$-coefficient in the second term.. Does anyone perhaps know a generalised version of Hopf-Cole with non-constant coefficients?
Thanks! Any hint would be helpful!