Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically?
Let $P(u,x)$ satisfy the following PDE:
$\frac{1}{2}\sigma_u^2 uP_{uu}+\frac{1}{2}\sigma^2_x x^2 P_{xx} - \theta_u (u-1)P_u +\mu_x xP_x - \rho P+u.x-f=0 $ for $u>\underline{u}(x)$
where $\underline{u}(x)$ is a free boundary to be determined.
Boundary conditions:
$$\lim_{x\rightarrow0} P(u,x) = 0$$
$$\lim_{x\rightarrow \infty} \frac{P(u,x)}{x} = a+b(u-1)$$
$$P(\underline{u}(x),x) = 0$$
$$P_u(\underline{u}(x),x) = 0$$