I am trying to solve an exercise in a stochastic calculus book, concerning stochastic optimal control. Unfortunately, I've been stuck for a day on a pesky PDE now.
Basically, I have reduced the optimization problem to the following PDE, to be solved on $[0,T]\times \mathbb{R} $, where $V(t,x)$ is the unknown function (i.e. HJB for value function):
$$V_t+e^{-\delta t} \ln\left(e^{-\delta t} V_x \right)-\Bigg(\frac{(\alpha-r)V_x}{x \sigma^2V_{xx}}(\alpha-r)x+(rx-e^{\delta t}V_x) \Bigg)V_x+\frac{1}{2}\frac{(\alpha-r)^2V_x^2}{\sigma^2V_{xx}}=0$$ with boundary value $V(T,x)=K \ln x$. To be perfectly clear, $r,\alpha, \sigma, \delta$ do not depend on $x$ nor $t$.
I have tried the following Ansätze: $V(t,x)=e^{-\delta t} g(x), \: V(t,x)=e^{-\delta t}\ln(x)f(t),\: V(t,x)=e^{-\delta t}f(t)g(x) $ but I always either end up with an annoying non-linearity of the form $\ln f$ or $e^f$ in the ODEs I arrive at, which makes the solution an inverse of a transcendetal integral, which simply is no fun (depending on whether I assume $f=e^h$ or not). I am pretty sure there should be an appropriate ansatz for this problem as it is hinted in the problem formulation that one should attempt such a solution. However my attemps have been to no avail.
I would appreciate any help or guidance. Thank you in advance.
It turns out I got lost when dealing with all the small terms. If one makes the ansatz $V(t,x)=f(t)\ln x + g(t)$ the problem simplifies nicely into an ODE.