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An Equidistant Truncated Euler--Maruyama Method for Nonlinear Time-Changed Stochastic Differential Equations

This paper proposes an equidistant truncated Euler–Maruyama method for solving nonlinear time-changed stochastic differential equations with superlinear growth, establishing a strong convergence rate arbitrarily close to $\min\{\beta_1, \alpha\beta_2, \alpha/2\}$ under polynomial growth, monotonicity, and Hölder regularity conditions.

Original authors: Ruchun Zuo, Qian Guo

Published 2026-09-21

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CC BY 4.0

Original authors: Ruchun Zuo, Qian Guo

Original paper licensed under CC BY 4.0 (https://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 a world where particles do not move in the smooth, predictable lines of a river, but instead get trapped in a chaotic maze, waiting for long, random intervals before they can take another step. This is the reality of "anomalous diffusion," a phenomenon observed in everything from the movement of proteins inside a living cell to the erratic fluctuations of financial markets. To describe this behavior, scientists use a special kind of mathematical model called a time-changed stochastic differential equation. Think of it as a standard equation for motion that has been hooked up to a broken clock. While the equation tries to move forward, the clock itself—governed by a random process—sometimes stalls or jumps unpredictably, causing the particle to freeze or leap forward in ways that standard physics cannot easily predict.

The challenge for mathematicians and computer scientists is that these equations rarely have neat, exact answers written down in a textbook. To understand how these systems behave, researchers must rely on computers to simulate them, breaking time into tiny steps and calculating the path forward. However, when the forces driving these particles grow very large or behave in complex, non-linear ways, standard simulation methods often fail. They can produce results that explode into infinity or drift wildly away from reality, rendering the simulation useless. This is particularly difficult when the "drift"—the force pushing the particle—depends on where the particle currently is, creating a feedback loop that is hard to control on a computer.

In a recent study, researchers Ruchun Zuo and Qian Guo from Shanghai Normal University tackled this specific problem by developing a new way to simulate these complex systems. They focused on a scenario where the driving force depends on the particle's current position, a situation that had previously caused existing computer methods to break down. The team created a new algorithm, which they call the equidistant truncated Euler–Maruyama method. The core idea is to force the computer to look at the system at perfectly even intervals of physical time, rather than letting the random, broken clock dictate the timing. To handle the dangerous, explosive growth of the forces, they added a safety mechanism: a "truncation" rule that gently caps the size of the forces if they get too large, preventing the simulation from spiraling out of control while still preserving the essential behavior of the system.

The researchers proved mathematically that this new method works reliably. They showed that as the time steps get smaller, the computer's approximation gets closer and closer to the true path of the particle. Crucially, they determined exactly how fast this improvement happens. The speed of convergence depends on three factors: how smoothly the physical forces change over time, how smoothly the random "clock" behaves, and the specific nature of the randomness itself. Their analysis revealed that the method can achieve a high level of accuracy, limited only by the smoothness of the underlying processes. In cases where the physical forces are smooth and the random clock behaves well, the method becomes highly efficient, offering a precise way to track these erratic movements.

To confirm their theory, the team ran a series of computer experiments. They tested the method on several different scenarios, including a simple linear case where the answer was known, and more complex cases where the forces grew rapidly and changed in irregular patterns. In every test, the new method performed as predicted. When they compared the computer's results to a highly accurate reference solution, the errors decreased at the exact rate their mathematics had foreseen. For instance, in a test involving a specific type of random time delay, the error dropped steadily as the time steps were refined, matching the theoretical slope perfectly. These simulations confirmed that the method is robust, capable of handling the superlinear growth and irregular time changes that had stumped previous approaches.

This work provides a solid foundation for simulating complex, real-world systems where particles get trapped and forces depend on position. By resolving the issue of state-dependent drift on a uniform time grid, the researchers have opened the door to more accurate models of biological transport, financial markets, and other systems governed by anomalous diffusion. Their method does not just offer a new tool; it provides a guaranteed way to measure how close a simulation is to the truth, ensuring that when scientists look into the chaotic behavior of these systems, the picture they see is reliable.

Technical Summary: An Equidistant Truncated Euler–Maruyama Method for Nonlinear Time-Changed Stochastic Differential Equations

Problem Statement
The paper addresses the numerical approximation of nonlinear time-changed stochastic differential equations (TCSDEs) driven by an inverse $\alpha$ -stable subordinator. Specifically, the study focuses on equations of the form:
$X(t) = X(0) + \int_0^t h(s, E(s), X(s)) ds + \int_0^t f(s, E(s), X(s)) dE(s) + \int_0^t g(s, E(s), X(s)) dB(E(s))$
where $h$ represents a physical-time drift, $f$ an operational-time drift, and $g$ a diffusion term driven by a time-changed Brownian motion $B(E)$ . The coefficients are allowed to exhibit superlinear growth in the state variable $X$ .

Existing numerical methods for TCSDEs predominantly rely on operational-time grids (D-grids), where time steps are defined by the subordinator $D$ . The authors note a critical limitation in this approach: operational grids strictly require the physical-time drift $h$ to be state-independent (typically of the form $h(E)$ ). When the physical drift depends on the state variable $X$ , the operational step size fails to control the "physical-time freezing error," rendering standard error analyses inapplicable. Furthermore, standard explicit Euler–Maruyama (EM) schemes diverge for equations with superlinear coefficients. While a physical-time equidistant framework was previously established by Zuo [34], that analysis was restricted to autonomous equations without physical-time drift and did not address the distinct regularities inherent to models with state-dependent physical drift.

Methodology
To overcome these limitations, the authors propose an Equidistant Truncated Euler–Maruyama (ETEM) method. The methodology involves three core components:

Equidistant Physical-Time Discretization: Unlike previous works that map equidistant operational steps to physical time, this scheme fixes a uniform grid in physical time $t_n = n\Delta$ . The numerical solution is advanced using the random increments of the inverse subordinator $E(t_{n+1}) - E(t_n)$ and the Brownian motion $B(E(t_{n+1})) - B(E(t_n))$ . This allows for the direct handling of state-dependent physical drifts $h(t, E(t), X(t))$ .
State Truncation: To manage the superlinear growth of coefficients, the authors employ a truncation mechanism. A truncation radius $R_\Delta = \Delta^{-\theta/(\ell+1)}$ is defined, where $\ell$ is the growth exponent. The coefficients are modified by projecting the state variable onto a closed ball of radius $R_\Delta$ before evaluation. This ensures the numerical scheme remains stable and moments do not explode in finite time.
Convergence Analysis: The analysis relies on establishing moment bounds for the inverse subordinator and utilizing generalized Gronwall inequalities adapted to the mixed time scales. The authors decompose the error into components arising from physical-time regularity, operational-time regularity, and the diffusion term.

Key Contributions
The paper makes two primary contributions:

Resolution of State-Dependent Physical Drift: The ETEM scheme extends the physical-time equidistant framework to rigorously handle TCSDEs where the physical-time drift $h$ depends on the state variable $X$ . This overcomes the structural limitation of operational-grid methods and extends previous physical-time grid analyses (which were restricted to drift-free models) to the nonautonomous case with state-dependent drift.
Strong Convergence Rate: Under assumptions of polynomial growth, local monotonicity, and Hölder regularity (with exponents $\beta_1$ in physical time and $\beta_2$ in operational time), the authors prove that the ETEM scheme converges strongly in the $L^r$ path-supremum norm. The convergence rate is shown to be arbitrarily close to:
$\min\{\beta_1, \alpha\beta_2, \alpha/2\}$
where $\alpha$ is the stability index of the subordinator. This result explicitly quantifies how the regularity of the coefficients and the properties of the subordinator govern the numerical error.

Results
The theoretical findings are supported by numerical simulations on three test cases:

Linear TCSDE: A linear equation with multiplicative noise where the theoretical rate is $\min\{1, \alpha, \alpha/2\}$ . With $\alpha=0.6$ , the observed rate matches the theoretical prediction of $0.3$.
Nonautonomous Superlinear TCSDE: A polynomial growth model with $\alpha=0.8$ . The theoretical rate is $\min\{1, 0.8, 0.4\} = 0.4$ , which is confirmed by the numerical experiments.
Low Time Regularity TCSDE: A model featuring a chirp perturbation that reduces the physical-time Hölder regularity to $\beta_1 = 0.3$ . With $\alpha=0.9$ , the theoretical rate becomes $\min\{0.3, 0.9, 0.45\} = 0.3$ . The simulations confirm that the convergence rate is limited by the lowest regularity parameter, regardless of the other factors.

Significance
The paper claims that the ETEM method provides a robust framework for simulating complex anomalous diffusion processes where system dynamics involve both deterministic state-dependent drifts and anomalous trapping. By establishing a strong convergence rate that adapts to the specific regularities of the physical and operational time scales, the method offers a practical tool for models in biological transport, financial markets, and disordered media where previous operational-grid methods were structurally inadequate. The work bridges the gap between the theoretical requirements of state-dependent drifts and the practical necessity of stable numerical schemes for superlinear TCSDEs.

Technical Summary: An Equidistant Truncated Euler–Maruyama Method for Nonlinear Time-Changed Stochastic Differential Equations

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