Two explicit numerical schemes are reported to keep numerical solutions of stochastic differential equations inside a prescribed multidimensional box at every grid point for any timestep, according to a new arXiv preprint. The result is conditional on the paper's stated assumptions.
The paper studies finite-dimensional Ito stochastic differential equations, or SDEs, driven by Brownian noise. Their permitted state space is a bounded hypercube - in plain terms, a multidimensional box - with arbitrary dimension, multidimensional noise and, where applicable, drift.
A hard boundary for a random system
These are explicit schemes, meaning the next numerical value is calculated directly from quantities available at the current step. The construction freezes the coefficient factor functions while leaving sigma variable, a feature the paper identifies as a key domain-preserving ingredient.
Under Assumptions 1 and 2, schemes (17) and (33) keep every numerical iterate in the hypercube almost surely when the initial value is in the hypercube, regardless of the timestep size. In ordinary language, "almost surely" means the guarantee can fail only on an event with probability zero.
That is a guarantee for the discrete points generated by the algorithms, within the SDE class covered by the paper. It is not a blanket statement that the same behavior holds for equations outside those conditions.
Accuracy comes with conditions
To describe accuracy, the paper uses two measures. Strong convergence is reported through mean-square error, while weak convergence concerns expected values of smooth test functions. For scheme (17), the stated orders are one-half for strong convergence and one for weak convergence.
An order here describes how the error scales as the timestep is reduced. The general scheme therefore combines domain preservation with a stated strong rate of one-half and a weak rate of one, subject to the theorem assumptions.
A separate result gives scheme (33) strong order one when the Brownian noise is one-dimensional, written M = 1. In that comparison, scheme (17) remains at one-half, and the higher-order theorem requires a sufficiently small timestep.
The qualification is important: the supplied analysis does not establish strong order one for multidimensional Brownian noise. The higher-order claim is limited to the one-noise case.
For Stratonovich equations, the authors construct variants by applying schemes (17) and (33) to the equivalent Ito formulation. Scheme (33) is used for the one-dimensional-noise case.
The evidence from simulation
The domain-preservation experiment used d = 2 and M = 1, so it tested a two-component system with one Brownian driver. It measured the proportion of independent realizations whose numerical solutions stayed in the hypercube at all grid times.
In the DP(0.5) cases, both proposed schemes retained both components in the domain for all 100 of 100 realizations in each of two cases. Euler-Maruyama retained 91 and 92 of 100 in the first case, and 90 and 14 of 100 in the second.
The DP(1.0) cases showed the same all-or-nothing result: both schemes retained both components for all 100 of 100 realizations in both cases. Milstein retained 99 and 100 of 100 in the first case, then 90 and 89 of 100 in the second.
Those results speak specifically to domain retention in the reported test cases. They do not test the arbitrary-dimensional setting in full, because the validation used two system components and one noise dimension, while the analyzed framework allows arbitrary dimension and multidimensional noise.
Separate convergence experiments used Monte Carlo averaging to estimate expectations, with 103 independent realizations for strong-error tests and 106 for weak-error tests, as reported in the supplied extraction. These simulations were used alongside the paper's theoretical statements about convergence rates.
A result for numerical methods
Taken together, the evidence is a methods result: the paper presents explicit schemes that preserve a bounded domain under its assumptions while retaining stated strong and weak convergence orders, with computational domain checks in a small test system. The evidence is mathematical and computational, centered on SDE schemes and simulated paths.
The supplied document is arXiv:2608.25685v1, dated 26 August 2026, and labeled a preprint. It reports support from SFVE-A, Swedish Research Council projects 2018-04443 and 2024-04536, and grant agreement 2022-06725 for computing resources.
Paper data and sources
Original title: Explicit domain preserving numerical schemes for a class of stochastic differential equations
Authors: Charles-Edouard Bréhier, David Cohen
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text