Summary: We consider the exact path sampling of the squared Bessel process and other continuous-time Markov processes, such as the Cox-Ingersoll-Ross model, the constant elasticity of variance diffusion model, and confluent hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale transformation, and change of measure. All these diffusions are broadly used in mathematical finance for modeling 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 the boundary of the state space. Such an approach allows us to simplify simulation schemes. The new methods are illustrated using pricing path-dependent options.