Before revision: This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of  non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model which is known as a semi-Markov modulated GBM model  .
After revision: This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model which is known as a semi-Markov modulated GBM model the locally risk minimizing price function satisfies a special case of this problem .