 Bessel process

For an integer $\delta \ge 1$, a Bessel process is the radial part $R_t=|W_t|$ of $\delta$-dimensional Brownian motion. The definition extends to every real $\delta \ge 0$ by taking the nonnegative square root of a squared Bessel process:Before its first hit of zero, the Itô formula givesThis singular equation describes the process away from zero; continuation at zero must be specified. At dimension one the process is Reflected Brownian motion and the global equation includes local time of a semimartingale. For dimensions below one the usual singular-drift equation cannot simply be asserted across zero. The squared Bessel process definition avoids that ambiguity.^[ref]