I am currently working on a regime-switching Black-Scholes model and am having trouble determining the existence and uniqueness of a solution for the problem. Specifically, I am interested in finding the solution to the regime-switching Black-Scholes equation with two regimes:
State 1: $$ \frac{\partial V_1}{\partial t}+\frac{1}{2} \sigma_1^2 S^2 \frac{\partial^2 V_1}{\partial S^2}+(r-\delta) S \frac{\partial V_1}{\partial S}-r V_1=\lambda_{12}\left(V_1-V_2\right) $$ with boundary conditions: $$ \begin{aligned} V_1(0, t) & =0 \\ \lim _{S \rightarrow \infty} V_1(S, t) & =S \\ V_1(S, T) & =\max \{S-E, 0\} . \end{aligned} $$ State 2: $$ \frac{\partial V_2}{\partial t}+\frac{1}{2} \sigma_2^2 S^2 \frac{\partial^2 V_2}{\partial S^2}+(r-\delta) S \frac{\partial V_2}{\partial S}-r V_2=\lambda_{21}\left(V_2-V_1\right) $$ with boundary conditions: $$ \begin{aligned} V_2(0, t) & =0 \\ \lim _{S \rightarrow \infty} V_2(S, t) & =S \\ V_2(S, T) & =\max \{S-E, 0\} . \end{aligned} $$
I am wondering what techniques can be used to determine the existence and uniqueness of a solution for this problem, and what conditions need to be satisfied for a solution to exist and be unique.
None of the references ((1),(2),(3)) provide us any thing about the existence and uniqueness of solutions for two-state regime switching option pricing models, while the generalized Black-Scholes models (described in page no : 1485, 779, 1768,2229) possess existence and uniqueness of solutions under reasonable regularity conditions.
I felt uncomfortable with the necessity and sensibility of numerical solutions for option pricing models (Equations (12) and (13)) under regime switching economy. Can we say the anything about the existence and uniqueness of such models?
Thank you for your help!