Exact Strong Approximation Rates for CIR Processes

Definitions and mathematical statement

Let $a,\sigma>0$ and $b,x\geq0$. Let $W$ be a standard real Brownian motion, and let $X$ be the nonnegative Cox--Ingersoll--Ross process satisfying \[ X_t=x+\int_0^t(a-bX_s)\,{\,\mathrm{d}} s+\int_0^t\sigma\sqrt{X_s}\,{\,\mathrm{d}} W_s. \] The stochastic integral is understood in the Itô sense. The parameter range includes $x=0$ and $b=0$; no condition excluding attainment of the boundary $0$ is imposed.

For an explicit interpretation of the approximation question, fix a deterministic terminal time $T>0$. For $N\in{\mathbb{N}}$, define the optimal absolute-mean error from equidistant Brownian observations by \[ e_N(T;a,b,\sigma,x) :=\inf_{\substack{\varphi:{\mathbb{R}}^N\to{\mathbb{R}}\\ \varphi\text{ Borel measurable}}} {\mathbb{E}}\!\left[\left|X_T-\varphi\left(W_{T/N},W_{2T/N},\ldots,W_T\right)\right|\right]. \] The approximation map may depend on the fixed parameters and on $N$. One precise convention for its optimal algebraic convergence order is \[ \rho(T;a,b,\sigma,x) :=\sup\left\{r\geq0:\sup_{N\geq1}N^r e_N(T;a,b,\sigma,x)<\infty\right\}. \] Under these explicit conventions, the question is whether, for every fixed $T>0$, $a,\sigma>0$, and $b,x\geq0$, \[ \rho(T;a,b,\sigma,x) =\min\left\{\frac{2a}{\sigma^2},1\right\}. \] Here the quantifiers concern each fixed parameter choice; constants in error estimates may depend on those parameters and on $T$.

Unresolved formulation qualification. The original statement refers to an approximation error in equation (4), but that equation is absent from the supplied record. The displayed information class and definition of algebraic order therefore make an interpretation explicit rather than reproduce a verified definition from the report. In particular, the record does not specify whether the intended claim is only the equality of algebraic orders above or the stronger assertion that, writing $r=\min\{2a/\sigma^2,1\}$, there exist constants $0<c\leq C<\infty$ and $N_0$ such that \[ cN^{-r}\leq e_N(T;a,b,\sigma,x)\leq CN^{-r} \qquad(N\geq N_0). \] These assertions are not interchangeable without further information. The positive terminal time and the exact observation grid also require confirmation against the missing equation.

Short English statement

The question asks whether the best mean absolute error for approximating a CIR process at a fixed positive time from $N$ equally spaced Brownian observations has algebraic order $\min\{2a/\sigma^2,1\}$ for every $a,\sigma>0$ and $b,x\geq0$. The missing equation (4) leaves the precise error class and meaning of exact order unresolved. The supplied record labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30003390$, source problem code catalogue entry, title Exact Strong Approximation Rates for CIR Processes. This draft uses the supplied complete registry record. No individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Stochastic Differential Equations: Regularity and Numerical Analysis in Finite and Infinite Dimensions (2018), Oberwolfach Reports, pp. 453--498, DOI 10.4171/owr/2017/9. https://doi.org/10.4171/owr/2017/9. The background describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The record separately gives proposed year 2017 and the tag owr-year:2017; the supplied citation's 2018 date is retained without reconciliation.
• Supplied literature source, "Oberwolfach report": https://ems.press/content/serial-article-files/46671.
• Supplied literature source, "Hefter--Herzwurm": https://doi.org/10.1016/j.jmaa.2017.10.076. No further bibliographic details are supplied.
• Supplied literature source, "Hefter--Jentzen": https://arxiv.org/abs/1710.08707. No further bibliographic details are supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and reports that the conjectured rate is proved in boundary and other parameter regimes, with matching lower bounds known in parts, while the full exact-rate statement is not established throughout the positive parameter range. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a check against the official Oberwolfach report. That verification is supplied provenance; no source visit or independent verification is asserted here.