I met with a linear final value problem (option pricing problem) $$u_t(t,x)+D(t,x)u_{xx}(t,x)+C(t,x)u_x(t,x)+R(t,x)u(t,x)=0,~(t,x) \in (0,T) \times (0,\infty),$$ with $u(T,x)=u_0(x)$.
For an accurate numerical solution for the above problem, I need to justify the existence and uniqueness of solution. But I am stuck at that point to make the clarity about regularity of the above PDE. Here, we assume that the Diffusion, Convection and Reaction coefficients ($D$, $C$ and $R$ respectively) are sufficiently smooth and bounded. To proceed, I want to clear some ideas in a broad sense:
Under what conditions on the functions $D,C$ and $R$, we can ensure the existence of a unique solution for the above problem for a smooth function $u_0(x)$ on $(0,\infty)$? Similiarly, Under what conditions on the functions $D,C$ and $R$, we can ensure the existence of a unique solution fora particular non-smooth continuous function $u_0(x)=\max\{x-E,0\}$ where $E>0$ is a fixed constant? What about the smoothness of solution?
In addition, I have boundary conditions $u(t,0)=0$ and $u \to x$ as $x \to \infty$. What effect the boundary conditions can make on the existence of solution?
How the existence and uniqueness is working on such a backward parabolic problem?