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Invariant measures of the stochastic theta method for stochastic differential equations with super-linearly growing coefficients

This paper establishes the existence, uniqueness, and convergence of the numerical invariant measure for the stochastic theta method applied to stochastic differential equations with super-linearly growing drift and diffusion coefficients, thereby extending previous results on the backward Euler-Maruyama method and related works to this broader class of problems.

Original authors: Xiaotong Li, Wei Liu, Wenjie Xiao

Published 2026-06-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiaotong Li, Wei Liu, Wenjie Xiao

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 long-term weather patterns of a chaotic, stormy ocean. In mathematics, this ocean is represented by Stochastic Differential Equations (SDEs). These equations describe how things change over time when they are pushed by both a steady force (like a wind) and random, unpredictable jolts (like sudden waves).

Sometimes, these equations get very tricky. The "wind" (drift) and the "waves" (diffusion) can grow incredibly fast—so fast that they are called super-linear. Think of it like a snowball rolling down a hill that not only gets bigger but starts growing exponentially faster the bigger it gets. Standard math tools often break or explode when trying to simulate these runaway systems.

This paper introduces a specific tool called the Stochastic Theta Method to handle these wild, fast-growing systems and find their Invariant Measure.

What is an "Invariant Measure"?

Imagine you are watching a crowded dance floor. At the start, everyone is scattered randomly. As the music plays, people move, bump into each other, and swirl around. Eventually, even though individuals are still moving, the overall shape of the crowd stops changing. You might see a dense cluster near the DJ and a sparse area near the exit, and this pattern stays the same hour after hour.

That stable, unchanging pattern of the crowd is the Invariant Measure. It's the "steady state" or the "long-term average behavior" of the system. The paper asks: Can we use a computer to accurately predict what this final crowd pattern looks like, even when the dancers are moving in wild, super-fast ways?

The Problem with Standard Tools

Usually, mathematicians use a method called the Euler-Maruyama method (like a simple step-by-step calculator) to simulate these systems. However, when the forces grow super-linearly (like our runaway snowball), this simple calculator often fails. It might produce numbers so huge they crash the computer, or it might miss the true long-term pattern entirely.

The Solution: The Stochastic Theta Method

The authors propose using the Stochastic Theta Method. Think of this as a "smart, cautious calculator."

The "Theta" Dial: This method has a dial (called $\theta$ $θ$ ) that you can turn.
- If you turn it to 0, it acts like the simple, risky calculator (Euler-Maruyama).
- If you turn it to 1, it acts like the Backward Euler method, which is very stable but computationally heavy.
- The authors focus on settings where the dial is between 0.5 and 1. This makes the method "implicit," meaning it looks ahead to the next step to ensure it doesn't take a step so big that it falls off the cliff.

What Did They Prove?

The paper provides two major guarantees for this "smart calculator":

Existence and Uniqueness: They proved that if you use this method, the computer simulation will eventually settle down into a single, unique pattern (the numerical invariant measure). It won't keep oscillating wildly or produce different patterns depending on where you started the simulation.
Convergence: They proved that the pattern found by the computer is actually very close to the true pattern of the real mathematical ocean. As you make the time steps smaller (like taking smaller, more careful steps), the computer's answer gets closer and closer to the truth.

The "Secret Sauce"

The authors had to be very careful. Previous studies could only handle cases where the "waves" (diffusion) grew slowly, even if the "wind" (drift) grew fast. This paper is special because it handles the case where both the wind and the waves grow super-fast.

To do this, they had to develop new mathematical "safety nets" (using specific inequalities and assumptions) to prove that the simulation wouldn't blow up, even when the forces were at their most extreme. They showed that as long as you keep your time steps small enough and set the "Theta" dial correctly, the system remains stable.

The Proof in the Lab

To show this works, the authors ran computer simulations on two examples:

A One-Dimensional Model: A single variable that tries to return to a center point but gets pushed away violently by its own size. The simulation showed that no matter where they started the "particle," it eventually settled into the same bell-shaped curve.
A Two-Dimensional Model: Two variables interacting with each other, both with super-fast growth. The simulation showed the particles clustering in a specific, stable region, proving the method works even in complex, multi-dimensional chaos.

Summary

In short, this paper says: "We have a new, robust way to simulate chaotic systems where forces grow explosively. We proved mathematically that this method finds the correct long-term pattern, and our computer tests confirm that it works."

They didn't claim this solves climate change or cures diseases directly; they simply built a stronger, more reliable mathematical bridge to cross the gap between chaotic, fast-growing equations and their stable, long-term solutions.

Technical Summary: Invariant Measures of the Stochastic Theta Method for SDEs with Super-Linearly Growing Coefficients

Problem Statement
Stochastic Differential Equations (SDEs) with invariant measures are critical in various fields, including population dynamics and generative diffusion models. However, explicit forms of these invariant measures are rarely obtainable, necessitating robust numerical approximations. A significant challenge arises when the coefficients of the SDEs grow super-linearly. While implicit methods like the backward Euler-Maruyama (EM) method are known to handle such growth, the stochastic theta method (which includes backward EM as a special case) had not been rigorously analyzed for invariant measures when both the drift and diffusion coefficients exhibit super-linear growth. Previous works, such as Jiang et al. [10], addressed super-linear drift but required linear growth for the diffusion coefficient. This paper addresses the gap in approximating invariant measures for SDEs where both coefficients grow super-linearly.

Methodology
The authors employ the stochastic theta method to discretize the Itô-type SDE:
$dX(t) = b(X(t))dt + g(X(t))dW(t)$
The numerical scheme is defined as:
$X_{k+1} = X_k + \theta b(X_{k+1})h + (1 - \theta)b(X_k)h + g(X_k)\Delta W_k$
where $\theta \in (1/2, 1]$ . The analysis relies on three core assumptions regarding the coefficients $b$ and $g$ :

Local Lipschitz Continuity with Super-linear Growth: The coefficients satisfy a local Lipschitz condition with a polynomial growth rate $q > 1$ .
Monotonicity Condition: A one-sided Lipschitz condition ensuring the stability of the difference between two solutions.
Coercivity Condition: A condition ensuring the existence of a Lyapunov function to bound the moments of the solution.

The proof strategy involves establishing two auxiliary lemmas:

Lemma 3.1: Proves the boundedness of the second moment ( $E|X_k|^2$ ) of the numerical solution under the coercivity condition and a specific stepsize constraint.
Lemma 3.2: Demonstrates the exponential mean-square convergence of two numerical solutions starting from different initial values, establishing the contractive property of the method.

These lemmas are then utilized to prove the existence and uniqueness of the numerical invariant measure via the tightness of probability measures and the contraction in the $p$ -Wasserstein distance. Finally, the convergence of the numerical invariant measure to the exact invariant measure is derived using the triangle inequality, combining the exponential convergence of the numerical scheme to its invariant measure with the strong convergence rate of the stochastic theta method.

Key Results

Existence and Uniqueness: The paper proves that for SDEs with super-linearly growing drift and diffusion coefficients, the stochastic theta method generates a unique numerical invariant measure, provided the stepsize $h$ is sufficiently small and $\theta > 1/2$ .
Convergence to Exact Measure: The numerical invariant measure converges to the exact invariant measure of the underlying SDE in the $p$ -Wasserstein distance ( $p \in (0, 1)$ ) with a convergence rate of $O(h^{p/2})$ .
Generalization: The results extend the findings of Jiang et al. [10] (which required linear diffusion growth) to the case of super-linear diffusion. Furthermore, since the backward EM method corresponds to $\theta=1$ , these results generalize the findings of Liu et al. [16] regarding the backward EM method to the broader class of stochastic theta methods.

Significance and Claims
The authors position this work as a non-trivial extension of existing literature. They explicitly state that techniques from previous works [10, 16] cannot be directly adapted because the stochastic theta method retains an explicit part when $\theta < 1$ , requiring careful handling of the super-linear diffusion term. The paper claims to provide a rigorous theoretical foundation for using the stochastic theta method in long-time simulations of SDEs with super-linear coefficients, a scenario common in complex physical and biological models.

Numerical Validation
Two numerical examples are provided to illustrate the theoretical results:

Example 4.1: A scalar mean-reverting model with super-linear drift ( $x^3$ ) and diffusion ( $x^2$ ). Simulations show the empirical density functions stabilizing over time and becoming indistinguishable regardless of initial values, supporting the existence and uniqueness of the invariant measure.
Example 4.2: A two-dimensional SDE system with super-linear coefficients. The results demonstrate the long-time stability of the method and the concentration of sample paths near the equilibrium area, further validating the convergence to an invariant measure.

Future Directions
The paper concludes by noting that the current analysis relies on second moments, which imposes a negative coefficient requirement on the linear term of the drift. The authors suggest that future research could explore using $p$ -th moments for $p \in (0, 1)$ to relax these conditions. Additionally, they propose investigating the stochastic theta method for SDEs with state-dependent switching, which could aid in sampling from finite mixture distributions relevant to machine learning.