  We consider the exact path sampling of the squared Bessel process and some
other continuous-time Markov processes, such as the CIR model, constant
elasticity of variance diffusion model, and hypergeometric diffusions, which
can all be obtained from a squared Bessel process by using a change of
variable, time and scale transformation, and/or change of measure. All these
diffusions are broadly used in mathematical finance for modelling asset prices,
market indices, and interest rates. We show how the probability distributions
of a squared Bessel bridge and a squared Bessel process with or without
absorption at zero are reduced to randomized gamma distributions. Moreover, for
absorbing stochastic processes, we develop a new bridge sampling technique
based on conditioning on the first hitting time at zero. Such an approach
allows us to simplify simulation schemes. New methods are illustrated with
pricing path-dependent options.
