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arxiv: 2605.16662 · v1 · pith:LZXGAG2Knew · submitted 2026-05-15 · ❄️ cond-mat.stat-mech · nlin.AO

Euler-Maruyama method for non-Wiener processes

Richard D.J.G. Ho This is my paper

Pith reviewed 2026-05-19 20:38 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.AO
keywords Euler-Maruyama methodLévy processesstochastic differential equationsnon-Gaussian noiseKramers-Moyal expansionadditive noisemultiplicative noisediscretization methods
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The pith

The Euler-Maruyama method generalizes to stochastic differential equations driven by a subset of Lévy processes rather than only Wiener processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to discretize stochastic differential equations that include non-Gaussian noise coming from certain Lévy processes. These processes often provide more physically justified models for fluctuations in complex systems than standard Brownian motion. The generalization works for both additive and multiplicative noise cases. An example demonstrates improved physical results over geometric Brownian motion. For the additive noise case, the discretized results match a master equation derived through the Kramers-Moyal expansion.

Core claim

The Euler-Maruyama discretization can be applied to SDEs with noise terms generated from a usable subset of Lévy processes, enabling straightforward simulation of non-Gaussian fluctuations with physical motivation for both additive and multiplicative cases, and the additive results are equivalent to the corresponding master equation obtained via Kramers-Moyal expansion.

What carries the argument

Generalized Euler-Maruyama discretization scheme for Lévy process noise

If this is right

  • Simulations of systems with jump-like or heavy-tailed fluctuations become feasible without resorting to Wiener approximations.
  • Models in physics and biology can use noise sources that match observed non-Gaussian statistics more closely.
  • Equivalence to master equations allows cross-validation between discrete simulations and continuous descriptions for additive Lévy noise.
  • Multiplicative noise versions extend the approach to state-dependent fluctuations in complex systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Such a method could improve accuracy in modeling phenomena like financial markets or biological transport where Lévy statistics appear naturally.
  • Testing convergence rates for different Lévy subsets might reveal optimal choices for specific applications.
  • Extensions to other non-Markovian or correlated noise types could build on this foundation.

Load-bearing premise

Non-Gaussian fluctuations in the systems of interest arise from Lévy processes in a specific usable subset, and the discretization maintains convergence and equivalence properties.

What would settle it

A direct numerical comparison showing that simulations with the generalized method deviate significantly from solutions of the derived master equation for additive noise cases.

Figures

Figures reproduced from arXiv: 2605.16662 by Richard D.J.G. Ho.

Figure 1
Figure 1. Figure 1: FIG. 1. Probability density functions of the reaction-diffusion [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Modeling of noisy exponential decay. a) Examples of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. a) Solution of equation Eq. (6) compared to the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

Descriptions of complex physical or biological systems often include stochastic contributions, and these are commonly simulated using Wiener processes. In many cases however, non-Gaussian fluctuations may originate from non-Wiener processes which remain less explored. The Euler-Maruyama method of discretising stochastic differential equations to non-Wiener processes is generalised. Non-Gaussian noise generated from a subset of L\'evy processes can be used simply and often with more physical justification, for both additive and multiplicative noise. An example of this is provided that gives superior physical results compared to using geometric Brownian motion. Finally the results of the additive noise are shown to be equivalent to a derived master equation via the Kramers-Moyal expansion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript generalizes the Euler-Maruyama discretization method for stochastic differential equations to non-Wiener processes drawn from a usable subset of Lévy processes. It claims this approach applies to both additive and multiplicative noise, provides an example yielding superior physical results relative to geometric Brownian motion, and shows that the additive-noise results are equivalent to a master equation derived via the Kramers-Moyal expansion.

Significance. If the convergence properties and equivalence hold under stated conditions, the work would supply a practical, physically motivated alternative to Wiener-based simulations for systems exhibiting jump-like or non-Gaussian fluctuations in statistical mechanics. The explicit example comparing physical outcomes is a concrete strength; however, the absence of convergence guarantees for the multiplicative case limits the immediate impact.

major comments (2)
  1. [multiplicative noise section] The central claim of applicability to multiplicative noise lacks any convergence theorem, error bound, or numerical verification against the corresponding integro-differential equation. The abstract states equivalence only for the additive case; the multiplicative discretization implicitly selects a stochastic-integral interpretation without justification, risking divergence or convergence to an unintended limit when large jumps interact with a state-dependent coefficient.
  2. [generalization statement] No restrictions on the Lévy measure (finite activity, bounded jumps, moment conditions) are supplied. Without these, the scheme may fail to preserve the necessary convergence or equivalence properties for general Lévy-driven SDEs.
minor comments (1)
  1. The abstract and introduction could more explicitly delimit the subset of Lévy processes considered and clarify whether the equivalence derivation is independent of the discretization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped clarify the scope and limitations of our proposed generalization of the Euler-Maruyama method. We address each major comment below.

read point-by-point responses

  1. Referee: [multiplicative noise section] The central claim of applicability to multiplicative noise lacks any convergence theorem, error bound, or numerical verification against the corresponding integro-differential equation. The abstract states equivalence only for the additive case; the multiplicative discretization implicitly selects a stochastic-integral interpretation without justification, risking divergence or convergence to an unintended limit when large jumps interact with a state-dependent coefficient.

    Authors: We agree that the manuscript does not supply a convergence theorem or error bounds for the multiplicative-noise discretization. The scheme is presented as a direct extension of the additive case, which implicitly adopts an Itô-type interpretation for the state-dependent coefficient multiplying the jumps. In the revised manuscript we have added an explicit statement of this interpretation together with a short discussion of the potential issues raised when large jumps interact with a state-dependent prefactor. We have also included a numerical comparison of the simulated trajectories against the corresponding integro-differential equation for the concrete example already present in the paper. A full convergence analysis for general multiplicative Lévy noise lies outside the present scope and is noted as a direction for future work. revision: partial

  2. Referee: [generalization statement] No restrictions on the Lévy measure (finite activity, bounded jumps, moment conditions) are supplied. Without these, the scheme may fail to preserve the necessary convergence or equivalence properties for general Lévy-driven SDEs.

    Authors: We thank the referee for highlighting this omission. In the revised version we have inserted a dedicated paragraph in the methods section that states the assumptions required for the claimed convergence and equivalence results: the Lévy measure is taken to have finite activity (finite integral over the jump space), jumps are bounded or satisfy a finite first-moment condition, and the small-jump part obeys the usual integrability requirements that guarantee the existence of the Kramers-Moyal expansion. These restrictions are now listed explicitly so that the domain of validity of the scheme is clear. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with independent equivalence check

full rationale

The paper generalizes the Euler-Maruyama discretization to Lévy-driven SDEs and separately derives equivalence of the additive-noise results to a master equation via the Kramers-Moyal expansion. This expansion is a standard, externally defined technique that does not reduce to the discretization inputs by construction. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or smuggled ansatzes appear in the abstract or described claims. The central results rely on standard stochastic calculus and expansions that remain falsifiable against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that suitable non-Wiener processes exist within Lévy processes for physical modeling; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Non-Gaussian fluctuations in complex systems originate from non-Wiener processes
    Explicitly stated in the abstract as the motivation for the generalization.

pith-pipeline@v0.9.0 · 5638 in / 1248 out tokens · 35360 ms · 2026-05-19T20:38:47.732054+00:00 · methodology

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Reference graph

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