Optimal Strong Approximation of the One-dimensional Squared {B}essel Process
For researchers in numerical SDEs, this work reveals that adaption can yield infinitely faster convergence for a specific CIR process with accessible boundary, though the result is incremental as it focuses on a particular SDE.
The paper studies strong approximation of the one-dimensional squared Bessel process at a final time point, showing that adaptive algorithms achieve an infinite polynomial convergence rate while equidistant grid methods achieve only rate 1/2, demonstrating that adaption dramatically improves convergence.
We consider the one-dimensional squared Bessel process given by the stochastic differential equation (SDE) \begin{align*} dX_t = 1\,dt + 2\sqrt{X_t}\,dW_t, \quad X_0=x_0, \quad t\in[0,1], \end{align*} and study strong (pathwise) approximation of the solution $X$ at the final time point $t=1$. This SDE is a particular instance of a Cox-Ingersoll-Ross (CIR) process where the boundary point zero is accessible. We consider numerical methods that have access to values of the driving Brownian motion $W$ at a finite number of time points. We show that the polynomial convergence rate of the $n$-th minimal errors for the class of adaptive algorithms as well as for the class of algorithms that rely on equidistant grids are equal to infinity and $1/2$, respectively. This shows that adaption results in a tremendously improved convergence rate. As a by-product, we obtain that the parameters appearing in the CIR process affect the convergence rate of strong approximation.