# Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.

Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf Let $\mathcal{L}V=\displaystyle\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} +rS\displaystyle\frac{\partial V}{\partial S} +\displaystyle S\frac{\partial V}{\partial A}-rV$

Where $r$ and $\sigma$ are positive real numbers. Let $\Omega$ be an unbounded subset of $\mathbb{R}^2$. Consider the boundary value problem:

$$\left\{ \begin{array}{lll} \mathcal{L}V-\frac{\partial V}{\partial t}=0, &\;\text{on}\;\; \Omega\times(0,T),\\ V(S,A,0)=V_{0}(S,A)&\;\text{on}\; (S,A)\in \Omega,\\ V(S,A,t)=h(S,A,t)&\;\text{on}\; (S,A,t)\in\partial\Omega\times[0,T]. \end{array} \right.$$ where $V_0$ and $h$ satisfy the compatibility condition:

$$V_{0}(S,A)=h(S,A,0), \;\forall (S,A)\in\partial\Omega.$$ My question is: under what conditions the above system would have a unique solution $V$. What about the regularity of $V$ ? what would happen if $r$ is a function of $S$, i.e, r=r(S)? I have been searching for this result but I have not found one. If some one know, could you please let me know ? Thanks for your time.

• What are $i$, $\mathcal{L}_i$ and $V_i$? – Andrew Nov 4 '14 at 7:36