Suppose that we have the following time-dependent partial differential equation:
\begin{equation} \frac{\partial V(t, x)}{\partial t} = \frac{1}{2}\sigma^2 x\frac{\partial^2 V(t, x)}{\partial x^2}+ \theta(m-x)\frac{\partial V(t,x)}{\partial x} - wxV(t, x), \quad t> 0, x\in \mathbb{R} \\ V(0, x ) = f(x) = e^{-ux}, \quad x\in \mathbb{R} \end{equation}
where $\theta > 0$, $m>0$, $\sigma >0$, $u\in \mathbb{R}$, and $w\in \mathbb{R}$ are constants.
How can I come up with $\tilde{V}(t, x)$ as an initial solution for the above partial differential equation system? Could you please give me the general instruction on how I should proceed to find such a solution from the above partial differential equation?
