to appear in Contemporary Mathematics


We discuss a generalization of the well known squared Bessel process with real nonnegative parameter $\delta$ by introducing a predictable almost everywhere positive process $\gamma(t,\omega)$ into the drift and diffusion terms. The resulting generalized process is nonnegative with instantaneous reflection at zero when $\delta$ is positive. When $\delta$ is a positive integer, the process can be constructed from $\delta$-dimensional Brownian motion. In particular, we consider $\gamma_t = X_{t-\tau}$ which makes the process a solution of a stochastic delay differential equation with a discrete delay. The solutions of these equations are constructed in successive steps on time intervals of length $\tau$. We prove that if $ 0 < \delta < 2$, zero is an accessible boundary and the process is instantaneously reflecting at zero. If $\delta \leq 2$, $\liminf_{t\rightarrow\infty} X_t = 0$. Zero is inaccessible if $\delta \geq 2$.