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Splitting schemes and estimators for stochastic differential equations with Hölder multiplicative noise

This paper introduces the first explicit pseudo-likelihood estimators based on Lie-Trotter and Strang splitting schemes for univariate stochastic differential equations with Hölder continuous multiplicative noise, proving their strong mean-square convergence, state space preservation, and superior accuracy and efficiency compared to existing Euler-Maruyama-based methods.

Original authors: Bowen Fang, Dario Spanò, Massimiliano Tamborrino

Published 2026-05-19

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Bowen Fang, Dario Spanò, Massimiliano Tamborrino

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the path of a tiny, jittery particle moving through a fluid. This particle is being pushed by a steady wind (the "drift") but is also being kicked randomly by invisible molecules (the "noise"). In the world of mathematics, this is called a Stochastic Differential Equation (SDE).

The problem is that for many real-world scenarios—like how interest rates change, how genes spread in a population, or how a neuron fires—the "kicks" from the molecules aren't uniform. They get stronger or weaker depending on where the particle is. This is called multiplicative noise, and it makes the math very messy.

Here is what the authors of this paper did, explained simply:

The Problem: The "Broken Ruler"

To predict where the particle will be tomorrow, scientists usually use a method called Euler-Maruyama. Think of this like trying to draw a smooth, curvy road by connecting a series of straight lines.

The Issue: If the road curves sharply or the ground gets slippery (which happens with this specific type of noise), the straight lines might shoot off the road entirely. They might predict the particle is in a place where it physically cannot exist (like a negative interest rate or a negative population count).
The Consequence: Existing methods either break the rules of physics (predicting impossible values) or require such tiny, slow steps to stay accurate that they take forever to compute.

The Solution: The "Splitting" Strategy

The authors invented a new way to solve this puzzle called Splitting Schemes.

Imagine you are trying to navigate a complex maze with two types of obstacles:

The Wind: A steady force pushing you in a specific direction.
The Bumps: Random, jumpy terrain.

Instead of trying to solve the whole maze at once (which is hard and error-prone), the authors' method splits the problem into two separate, easy-to-solve steps:

Step A: Ignore the bumps for a moment and just calculate where the wind would take you. This is easy because the wind is predictable.
Step B: Now, take that new position and apply the random bumps. Because of a clever mathematical trick (called the Lamperti transform), the authors found a way to make these bumps behave like a simple, standard random walk, which is also easy to solve.

Finally, they stitch these two solutions back together.

The Two New Tools: LT and Strang

The paper introduces two specific ways to stitch the steps together:

Lie-Trotter (LT): You do Step A, then Step B. (Wind, then Bumps).
Strang (S): You do half of Step A, then all of Step B, then the other half of Step A. (Half-Wind, Bumps, Half-Wind).

Think of Strang like a chef who tastes the sauce halfway through cooking, adjusts the seasoning, and then finishes the dish. It's a bit more work, but the result is much more balanced and accurate.

Why This Matters

The authors proved that their new "Splitting" methods have three superpowers that the old methods lack:

They Stay on the Road: They never predict impossible values. If the particle is supposed to stay positive (like a population count), the math guarantees it stays positive.
They Are Stronger: They stay accurate even when you take big steps. You don't need to slow down to a crawl to get a good answer.
They Are Better Guessers: When the authors used these methods to estimate the hidden parameters of the system (like "how strong is the wind?" or "how bumpy is the road?"), their guesses were much more accurate and faster to compute than the current best methods.

The "Real-World" Tests

The team tested their idea on famous mathematical models used in finance and biology, such as:

The CIR Model: Used for interest rates.
The Wright-Fisher Model: Used for genetics.
The Ginzburg-Landau Equation: Used in physics.

In every test, their new "Splitting" methods outperformed the old ones. They were more accurate, stayed within the correct boundaries, and were computationally efficient.

The Bottom Line

The authors didn't just tweak an old formula; they built a new engine. By breaking a complex, jittery problem into two simple, solvable pieces and reassembling them carefully, they created a tool that is more robust, faster, and more reliable for understanding systems that move randomly but follow specific rules.

Technical Summary: Splitting schemes and estimators for stochastic differential equations with Hölder multiplicative noise

Problem Statement
The paper addresses the challenge of parameter estimation for univariate stochastic differential equations (SDEs) of the form $dX_t = f(X_t; \theta)dt + g(X_t; \sigma)dW_t$ , where the drift coefficient $f$ is locally Lipschitz and the diffusion coefficient $g$ is Hölder continuous with exponent $q \in (0, 1/2)$ . This class of models, which includes widely used processes such as the Cox-Ingersoll-Ross (CIR), Wright-Fisher, and Ahn-Gao models, presents significant difficulties for existing inference methods.

Standard approaches, such as the Euler–Maruyama (EuM) discretization, often fail to preserve the state space (e.g., failing to keep the CIR process non-negative) and lack strong mean-square convergence when coefficients are not globally Lipschitz. Alternative methods relying on Gaussian approximations or Hermite polynomial expansions (e.g., Aït-Sahalia's method) can suffer from computational inefficiency or instability with large observation time steps. Furthermore, existing splitting schemes have primarily been developed for SDEs with globally Lipschitz drifts or additive noise, leaving a gap for schemes that are both strong mean-square convergent and state-space preserving for SDEs with Hölder multiplicative noise.

Methodology
The authors propose a novel framework based on numerical splitting schemes derived from a specific decomposition of the original SDE. The methodology proceeds as follows:

SDE Decomposition: The authors exploit the concept of "reducible SDEs" and the Lamperti transform. They decompose the drift $f(x)$ by adding and subtracting the term $\frac{1}{2}g(x)g'(x)$ . This splits the original SDE into two sub-equations:
- An Ordinary Differential Equation (ODE): $dX^{[1]}_t = [f(X^{[1]}_t) - \frac{1}{2}g(X^{[1]}_t)g'(X^{[1]}_t)]dt$ .
- A solvable SDE: $dX^{[2]}_t = \frac{1}{2}g(X^{[2]}_t)g'(X^{[2]}_t)dt + g(X^{[2]}_t)dW_t$ .
  The second sub-equation is explicitly solvable via the Lamperti transform $Y_t = \int^x \frac{1}{g(u)}du$ , which reduces it to a process with unit diffusion.
Splitting Schemes: The exact solutions of these sub-equations are composed to approximate the solution of the original SDE. Two specific compositions are introduced:
- Lie-Trotter (LT) Scheme: Composes the solutions as $\phi^{[2]}_h \circ \phi^{[1]}_h$ .
- Strang (S) Scheme: Composes the solutions as $\phi^{[1]}_{h/2} \circ \phi^{[2]}_{h/2} \circ \phi^{[1]}_{h/2}$ .
  These schemes yield explicit, state-space preserving numerical approximations.
Pseudo-Likelihood Estimation: The authors derive explicit pseudo-likelihood functions based on the transition densities of these splitting schemes. Unlike Gaussian approximations, these densities are non-linear transformations of Gaussian variables. The negative pseudo-log-likelihoods are minimized to obtain estimators for the parameter vector $\alpha = (\theta, \sigma)^T$ .
Theoretical Analysis: The paper provides rigorous proofs for:
- Strong Mean-Square Convergence: Proving that the LT scheme converges strongly to the true solution even with Hölder diffusion coefficients, a result that requires ad-hoc proofs as standard Milstein-type theorems do not apply.
- One-Step Consistency: Establishing that the LT scheme has a one-step consistency of order $O(h^2)$ and the Strang scheme has order $O(h^3)$ .
- Asymptotic Properties: Proving the consistency and asymptotic normality of the LT estimator. A key technical challenge addressed is the "tangled" dependence of the pseudo-likelihood on both drift ( $\theta$ ) and diffusion ( $\sigma$ ) parameters, which necessitates new proof techniques distinct from those used for estimators with separable parameters.

Key Results

Numerical Performance: Extensive simulations on the CIR, Student diffusion, and Ahn-Gao models demonstrate that the proposed splitting schemes achieve strong mean-square convergence of order 1. The Strang scheme consistently outperforms the Lie-Trotter scheme and other competitors (EuM, Milstein, semi-discrete, Lamperti+EuM) in terms of mean-square error and Wasserstein distance to the true invariant density.
State Space Preservation: The schemes successfully preserve the state space (e.g., non-negativity for CIR, boundedness for Wright-Fisher) for entrance boundaries, a property often violated by EuM and Milstein schemes.
Estimation Accuracy: The proposed pseudo-maximum likelihood estimators (PMLE) based on the splitting schemes outperform existing benchmarks (Kessler, Hermite, Lamperti-based methods) in terms of accuracy, particularly for large observation time steps ( $h_{obs}$ ). The Strang estimator generally provides the most accurate inference when the underlying mapping is invertible; otherwise, the Lie-Trotter estimator remains robust.
Computational Efficiency: While the Strang scheme is slightly more computationally expensive than Lie-Trotter, both are significantly faster than Hermite expansion-based methods and comparable to or faster than Kessler's approach in many scenarios.

Significance and Claims
The paper claims to introduce the first explicit pseudo-likelihood estimators based on numerical splitting schemes that are simultaneously strong mean-square convergent and state-space preserving for the broad class of SDEs with locally Lipschitz drift and Hölder continuous multiplicative diffusion.

The significance of the work lies in:

Bridging a Gap: It fills the methodological void for inference in SDEs with Hölder noise, where previous methods either lacked convergence guarantees or failed to preserve essential structural properties of the solution.
Novel Decomposition: The specific decomposition strategy, which integrates the Lamperti transform directly into the splitting of the drift, allows for the exact solution of the multiplicative noise sub-equation, a task traditionally deemed difficult for splitting schemes in this context.
Theoretical Advancement: The derivation of consistency and asymptotic normality for estimators where drift and diffusion parameters are jointly entangled in the likelihood function represents a significant theoretical contribution, requiring new moment bounds and proof techniques.
Practical Utility: The proposed estimators offer a robust alternative for simulation-based inference and parameter estimation in complex financial, biological, and physical models (e.g., interest rate modeling, population genetics) where standard discretization methods fail.

The authors note that while the current work focuses on scalar SDEs, the approach is naturally extendable to multivariate settings via multivariate reducible SDEs and the multivariate Lamperti transformation. They also acknowledge that for certain models (like the Ahn-Gao model with large time steps), the invertibility required for the Strang estimator's likelihood may fail, in which case the Lie-Trotter estimator serves as a robust alternative.