## The Annals of Probability

- Ann. Probab.
- Volume 23, Number 4 (1995), 1875-1894.

### Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift

Denis R. Bell and Salah-Eldin A. Mohammed

#### Abstract

We establish the existence of smooth densities for solutions of $\mathbf{R}^d$-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 $\mathbf{R}^d$ and $g$ is an $n \times d$ matrix-valued function defined on $\lbrack 0,\infty) \times \mathbf{R}^d$, such that $gg^\ast$ has degeneracies of polynomial order on a hypersurface in $\mathbf{R}^d$. 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.

#### Article information

**Source**

Ann. Probab. Volume 23, Number 4 (1995), 1875-1894.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176987807

**Digital Object Identifier**

doi:10.1214/aop/1176987807

**Mathematical Reviews number (MathSciNet)**

MR1379172

**Zentralblatt MATH identifier**

0852.60063

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 34K50: Stochastic functional-differential equations [See also , 60Hxx] 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

**Keywords**

Hereditary delay systems Malliavin calculus smooth densities

#### Citation

Bell, Denis R.; Mohammed, Salah-Eldin A. Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift. Ann. Probab. 23 (1995), no. 4, 1875--1894. doi:10.1214/aop/1176987807. http://projecteuclid.org/euclid.aop/1176987807.