Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift

Authors

Denis R. Bell, University of North Florida
Salah-Eldin A. Mohammed, Southern Illinois University CarbondaleFollow

Comments

Published in Annals of Probability, 23(4), 1875-1894.

Abstract

We establish the existence of smooth densities for solutions of R^d-valued stochastic hereditary differential systems of the form

dx(t) = H(t,x)dt + g(t, x(t - r))dW(t).

In the above equation, W is an n-dimensional Wiener process, r is a positive time delay, H is a nonanticipating functional defined on the space of paths in R^d and g is an n x d matrix-valued function defined on [0, ∞) x R^d, such that gg^* has degeneracies of polynomial order on a hypersurface in R^d. In the course of proving this result, we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent principal part.

Recommended Citation

Bell, Denis R. and Mohammed, Salah-Eldin A. "Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift." (Jan 1995).