This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of non-local parabolic partial differential equation(PDE) which naturally arises in the study of derivative pricing in a generalized market model which is known as a semi-Markov modulated geometric Brownian motion(GBM) model. We study the well-posedness of the problem via a Volterra integral equation of second kind. A probabilistic approach, in particular the method of conditioning on stopping times is used for showing the uniqueness.