Optimal Strong Approximation of the One-dimensional Squared Bessel\n Process

2016/01/07 by Mario Hefter, Hefter, Mario, André Herzwurm +1
Decision Sciences · Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Numerical Analysis (math.NA) #Probability (math.PR) #Stochastic processes and financial applications #demographic modeling and climate adaptation

paper · pdf · doi:10.48550/arxiv.1601.01455

openalex publication_date 2016/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the one-dimensional squared Bessel process given by the\nstochastic differential equation (SDE) \dX_t = 1
,dt +\n2
sqrtX_t
,dW_t,
quad X₀=x₀,
quad t
in[0,1], and study\nstrong (pathwise) approximation of the solution X at the final time point\nt=1. This SDE is a particular instance of a Cox-Ingersoll-Ross (CIR) process\nwhere the boundary point zero is accessible. We consider numerical methods that\nhave access to values of the driving Brownian motion W at a finite number of\ntime points. We show that the polynomial convergence rate of the n-th minimal\nerrors for the class of adaptive algorithms as well as for the class of\nalgorithms that rely on equidistant grids are equal to infinity and 1/2,\nrespectively. This shows that adaption results in a tremendously improved\nconvergence rate. As a by-product, we obtain that the parameters appearing in\nthe CIR process affect the convergence rate of strong approximation.\n

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