Cox-Ingersoll-Ross (CIR) processes are essential in mathematical finance, serving as models for interest rates and volatility in various financial frameworks. These processes are unique strong solutions to specific scalar stochastic differential equations (SDEs) and maintain nonnegative values. Their study is particularly relevant for strong (pathwise) approximation due to the square root function's lack of smoothness in the diffusion coefficient. This thesis investigates strong approximation within a subclass of CIR processes that have a positive probability of hitting the boundary point zero, establishing strong convergence rates. We concentrate on the one-dimensional squared Bessel process at a single time point, comparing adaptive and nonadaptive algorithms. Nonadaptive algorithms utilize a fixed discretization of the driving Brownian motion, while adaptive algorithms can sequentially select evaluation points, such as those with step size control. Common nonadaptive methods include Euler- and Milstein-type approaches based on equidistant grids. We demonstrate that numerical algorithms relying solely on equidistant evaluations can achieve a maximum convergence order of 1/2. In contrast, we develop an adaptive algorithm that selects evaluation sites based on the path of the Brownian motion, proving its convergence at an arbitrarily high polynomial rate.
