A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients

TL;DR

This paper develops a Schauder approach to solve degenerate-parabolic PDEs with unbounded coefficients on a half-space, ensuring existence and uniqueness of solutions in weighted Holder spaces, with applications to probability and finance.

Contribution

It introduces a new method for establishing well-posedness of degenerate PDEs with unbounded coefficients using weighted Schauder spaces.

Findings

Proved existence and uniqueness of solutions in weighted Holder spaces.

Applied results to show well-posedness of associated martingale problems.

Extended classical PDE theory to degenerate, unbounded coefficient cases.

Abstract

Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half-space. We establish existence and uniqueness of solutions in weighted Holder spaces which incorporate both the degeneracy at the boundary and the unboundedness of the coefficients. In our companion article [arXiv:1211.4636], we apply the main result of this article to show that the martingale problem associated with a degenerate-elliptic partial differential operator is well-posed in the sense of Stroock and Varadhan.