A new numerical method for a class of scalar stochastic differential equations keeps its computed values nonnegative—never below zero—at every grid point, regardless of the time-step size, when it starts from a nonnegative value. The analysis also reports first-order strong convergence when the time-steps are sufficiently small. Here, “strong convergence” refers to the paper’s analysis of the error between the numerical and exact solutions.
The work asks whether an explicit scheme for scalar Itô SDEs can combine that sign guarantee with first-order strong convergence, and whether a corresponding construction works for Stratonovich SDEs. The supplied document is an arXiv version 1 preprint dated 26 August 2026.
A state constraint built into the calculation
The theoretical setup starts with a deterministic initial value x0 that is nonnegative. Under the stated coefficient assumptions, the exact scalar SDE has a unique solution that remains nonnegative almost surely for all times. That baseline matters because the numerical result is designed to retain the same lower bound at the discrete grid points.
The construction factors the drift and diffusion coefficients into the state variable and auxiliary mappings. That factorization is the basis for the explicit scheme analyzed in the paper. The question, and therefore the main guarantee, is framed for scalar SDEs under the stated coefficient conditions.
The guarantees split at the time-step
The clearest divide in the results is between the guarantee about the sign and the guarantee about accuracy. The nonnegativity theorem holds almost surely at every grid point for any time-step size, provided the initial value is nonnegative. The order-one strong-convergence theorem is narrower: it requires the stated assumptions and sufficiently small time-steps. A large step therefore remains covered by the sign statement, but not by the theorem’s first-order accuracy condition.
To prove convergence, the authors introduce a continuous-time auxiliary process that matches the numerical solution at the grid points. They derive bounds for its moments and increments and use those estimates in the strong-error analysis. The paper also derives pathwise strong and almost-sure convergence at rate 1 minus epsilon for any epsilon in (0,1).
The Stratonovich extension follows after Itô–Stratonovich conversion and requires additional regularity. With that extra regularity, the adapted Stratonovich scheme also has first-order strong convergence. The extension therefore carries a further condition beyond the basic statement for the Itô scheme.
What the simulations showed
The separate positivity experiment used 100 simulated realizations, with x0=0.1 and a time horizon of T=50. It counted the realizations that stayed positive at all times. The test covered listed coefficient choices, so it serves as a numerical check of the reported construction alongside the theorem.
Across those coefficient choices, PP(1.0) and PP(0.5) each produced 100 positive realizations out of 100. Milstein’s count ranged from 0 to 99, while Euler–Maruyama’s ranged from 0 to 10. The reported counts favored the two positivity-preserving variants in those tests; they are evidence about the listed simulations, not a result for every possible coefficient choice.
For convergence, the Itô demonstration used dyadic refinements, meaning repeated refinements of the time grid, and Monte Carlo averaging over independent simulated paths. Errors for the methods were compared with a reference generated by the proposed scheme at the finest step. That setup ties the numerical rate comparisons to a numerical reference rather than an independently known exact solution.
In the reported Itô examples, PP(1.0) and Milstein were observed to behave as first-order methods; PP(0.5) and Euler–Maruyama showed half-order strong convergence. In the reported Stratonovich experiment, PP(1.0), Euler–Heun and Milstein were observed to be first-order, while PP(0.5) showed half-order convergence. These observed rates belong to the reported setups.
A useful result with a narrow scope
Taken together, the results offer a particular combination: a sign guarantee that does not depend on the size of the time-step, plus a first-order strong-convergence guarantee that does depend on sufficiently small steps. The distinction is important when interpreting the method’s scope. The paper establishes nonnegativity, not strict positivity away from zero, and its theorems are conditional on the stated scalar coefficient class.
The Stratonovich result is likewise conditional on additional regularity after conversion. The central claims remain tied to the scalar setting and the coefficient assumptions named in the analysis.
The acknowledgements report support from SFVE-A, the Swedish Research Council, the European Union/ERC StochMan, and computing resources at NAISS and Chalmers e-Commons.
Paper data and sources
Original title: Analysis of a first-order explicit positivity preserving scheme for a class of scalar SDEs
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