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

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.

42 pages; corresponds to Part 1 of our previous article [arxiv.org:1112.4824v1]. Incorporates final galley proof corrections corresponding to published version

References in corpus (2)

• Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations
• On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Ito processes

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