Let's say I have a stochastic optimization problem of the type ($x \in \mathbb{R}^n$):
$$J(x,t)= \max_u E\bigg [\int_t^{+\infty} g(x,u)e^{-\rho s}ds\bigg]$$ The SHJB equation for this is ($J$ is the value function, $\mu (x,t,u)$ is the drift, I'm assuming constant $\sigma$ and $\nu = \sigma \sigma'$). $$\rho J = \max_{u} \bigg [ g + \langle \mu, \nabla J \rangle + \frac{1}{2}\sum_{ij}\frac{\partial^2 J}{\partial x_i \partial x_j}\nu_{ij}\bigg]$$ But with what boundary conditions am I supposed to solve this for infinite horizon? Even for the one dimensional case, is it a Cauchy problem? What's the initial condition?
Given I have 4 close votes for lack of context I really ought to add it. I'm looking for a way to approach this equation: https://mathoverflow.net/questions/467627/specific-type-of-pde
This is pretty much unmanageable in multiple dimensions (or isn't it?) but I was trying to solve it in one dimension (and $\rho = 0$): it's a Chini equation with non constant Chini invariant, so I was approaching it with a numerical solver. The shape remains pretty much the same if I vary the initial condition but the numerical values obviously don't.