Optimal Strong Approximation of the One-dimensional Squared {B}essel Process
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 S...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
07.01.2016
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| Subjects | |
| Online Access | Get full text |
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| Summary: | 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. |
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| DOI: | 10.48550/arxiv.1601.01455 |