Strong convergence rate of the fractional WIS Euler integrator
Fix $T > 0$, $H \in (0, 1)$, $\alpha, \beta, x_0 \in \mathbb{R}$ and a globally Lipschitz function $a\colon \mathbb{R} \to \mathbb{R}$ of at most linear growth. Let $B^H$ be fractional Brownian motion with $\mathbb{E}[B^H(t) B^H(s)] = \tfrac12 (t^{2H} + s^{2H} - |t - s|^{2H})$, and let $X$ solve $$dX(t) = [\alpha X(t) + a(X(t))]\,dt + \beta X(t)\,dB^H(t), \qquad X(0) = x_0,$$ with the stochastic integral in the Wick–Itô–Skorohod sense. For $n \ge 1$ set $h = T/n$, $t_i = ih$ and $$J_i = \exp\Bigl(-\alpha t_i + \frac{\beta^2}{2}\bigl[T^{2H} - (T - t_i)^{2H}\bigr] - \beta B^H(t_i)\Bigr), \qquad 0 \le i \le n,$$ and, with exact samples of the same path of $B^H$, $$Z_0 = x_0, \qquad Z_{i+1} = Z_i + h J_i\, a(Z_i / J_i), \qquad X_n = Z_n / J_n.$$ This is the GfBm_EM scheme of Erdoğan, Lord and Schieven at the terminal time, an Euler method for the random ordinary differential equation obtained from an integrating factor. For drifts satisfying these conditions (and, more generally, Hölder continuous in time) they proved $(\mathbb{E}|X(T) - X_n|^2)^{1/2} \le C h^{H}$, and their numerical experiments led them to conjecture the rate below for autonomous drifts.
Problem. Does every such fixed problem admit a finite constant $C$, independent of $n$, such that $$\bigl(\mathbb{E}|X(T) - X_n|^2\bigr)^{1/2} \le C h^{\min\{H + 1/2,\, 1\}}?$$ The constant may depend on all of the fixed data.
Source: Erdoğan, Lord and Schieven (2026), conjecture (1.3).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- PR
- Keywords
- AIM Open Applied Problems, fractional Brownian motion, strong convergence, Wick–Itô–Skorohod integral
References
@article{erdogan2026,
author = {Erdo{\u{g}}an, U. and Lord, G. J. and Schieven, R. B.},
title = {Numerical approximation of {SDE}s driven by fractional {B}rownian motion for all {$H \in (0,1)$} using {WIS} integration},
journal = {IMA J. Numer. Anal.},
year = {2026},
doi = {10.1093/imanum/drag055},
eprint = {2404.07013},
archivePrefix = {arXiv},
}
@misc{aim-wis-fractional-euler-rate,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Strong convergence rate of the fractional WIS Euler integrator}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/639-wis-fractional-euler-rate.md},
note = {Entry 639 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove a terminal strong convergence bound for the GfBm\_EM Euler integrator of a scalar stochastic differential equation driven by multiplicative fractional Brownian noise in the Wick--Itô--Skorohod sense. For every Hurst parameter \(H\in(0,1)\) and every autonomous globally Lipschitz drift, the root mean-square error is bounded by \(C h^{\min\{H+1/2,1\}}\), using exact samples of the same noise path. The drift need not …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00778, "Strong convergence rate of the fractional WIS Euler integrator", clopen.solutions.