Strong approximation of a Cox-Ingersoll-Ross process via approximation of the minimum of brownian motion

Cox-Ingersoll-Ross (CIR) processes are widely used in mathematical finance, e. g., as a model for interest rates or as the volatility process in the Heston model. CIR processes are unique strong solutions of particular scalar stochastic differential equations (SDEs) and take nonnegative values. These SDEs are of particular interest in the context of strong (pathwise) approximation due to the lack of smoothness of the square root function appearing in the diffusion coefficient of the SDE. In this thesis we study strong approximation for a particular subclass of CIR processes, where the corresponding processes hit the boundary point zero with positive probability, and prove strong convergence rates. We focus on strong approximation of the one-dimensional squared Bessel process at a single point in time with emphasis on the comparison of adaptive and nonadaptive algorithms. In the context of SDEs, a nonadaptive algorithm uses a fixed discretization of the driving Brownian motion whereas adaptive algorithms may sequentially choose the evaluation points as, e. g., algorithms with a step size control. The most frequently studied nonadaptive algorithms are Euler- and Milstein-type methods that are based on values of the driving Brownian motion on an equidistant grid. We prove that arbitrary numerical algorithms that only use point evaluations of the Brownian motion on equidistant grids achieve at most a convergence order 1/2 with respect to the number of point evaluations. On the other hand, we construct an adaptive algorithm, which sequentially chooses the evaluation sites of the Brownian motion in a path-dependent way, and prove its convergence at an arbitrarily high polynomial rate.