A system of non-local parabolic PDE and application to option pricing

TL;DR

This paper proves the well-posedness of a non-local degenerate parabolic PDE system arising in derivative pricing within a generalized market model, using probabilistic and integral equation methods.

Contribution

It establishes the existence and uniqueness of solutions for a complex PDE system in a financial mathematics context, extending previous models.

Findings

Well-posedness of the PDE system is proven.

The solution approach involves Volterra integral equations.

Probabilistic methods confirm uniqueness.

Abstract

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