On a Class of Semi-Elliptic Diffusion Models - Part I: A Constructive Analytical Approach for Global Solutions, Densities and Numerical Schemes with Applications to the LIBOR Market Model

Abstract

Semi-elliptic stochastic differential equations (SDEs) are common models among practitioners. However, the value functions and sensitivities of such models are described by degenerate parabolic partial differential equations (PDEs) where the existence of regular global solutions is not trivial, and where densities do not exist in spaces of measurable functions but only in a distributional sense in general. In this paper, we show that for a related class of such equations regular global solutions can be constructed. Moreover, the solution scheme has a probabilistic interpretation where the existence of regular densities on certain subspaces of the state space can be exploited. Prominent examples of models of practical interest belonging to this class include factor reduced LIBOR market models and Cheyette models. Moreover, factor reduced SDEs originating from a full factor model are in the class to which our theorem applies. The result is also of interest for the theory of degenerate parabolic equations. A more detailed analysis of numerical and computational issues, as well as quantitative experiments will be found in the second part.