Squared Bessel process with delay

Abstract

We discuss a generalization of the well known squared Bessel process with real nonnegative parameter δ by introducing a predictable almost everywhere positive process γ(t, ω) into the drift and diffusion terms. The resulting generalized process is nonnegative with instantaneous reflection at zero when δ is positive. When δ is a positive integer, the process can be constructed from δ-dimensional Brownian motion. In particular, we consider γt = Xt−τ 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 τ . We prove that if 0 < δ < 2, zero is an accessible boundary and the process is instantaneously reflecting at zero. If δ ≤ 2, lim inft→∞Xt = 0. Zero is inaccessible if δ ≥ 2.