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

Invariant Measures of the Stochastic Theta Method for SDEs with Super-Linear Coefficients

Overview

The paper investigates the theoretical properties of the stochastic theta method as a numerical scheme for approximating invariant measures of stochastic differential equations (SDEs) whose drift and diffusion coefficients exhibit super-linear growth. The analysis extends prior work on invariant measure approximation to cover cases where both drift and diffusion coefficients are not globally Lipschitz but rather grow faster than linearly. The existence, uniqueness, and convergence of the numerical invariant measure to the true invariant measure of the SDE are rigorously derived, and the generality of the results is underscored by their applicability to the Backward Euler-Maruyama method as a special case.

Mathematical Formulation and Assumptions

The study frames its analysis around the SDE of Itô-type:

$$dX(t) = b(X(t))\,dt + g(X(t))\,dW(t), \quad X(0) = x_0 \in \mathbb{R}^n$$

where $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is the drift and $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$ the diffusion, each allowed to grow super-linearly (i.e., their increments may scale as $|x|^q$ for some $q>1$). The stochastic theta method, parameterized by $\theta \in (1/2,1]$, interpolates between fully explicit ($\theta=0$, Euler-Maruyama) and fully implicit ($\theta=1$, Backward Euler-Maruyama) schemes:

$$X_{k+1} = X_k + \theta b(X_{k+1}) h + (1-\theta) b(X_k) h + g(X_k)\Delta W_k,\quad X_0 = x_0$$

where $h$ is the temporal discretization and $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$0 the Brownian increment.

Three crucial structural assumptions are enforced:

• Local polynomial Lipschitz continuity: For some $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$1, $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$2, $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$3.
• Dissipativity-type monotonicity ($b:\mathbb{R}^n \rightarrow \mathbb{R}^n$4): $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$5.
• Coercivity for moment growth control: $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$6.

The non-globally Lipschitz structure of $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$7 and the inclusion of super-linear growth for both coefficients present major challenges in both numerical simulation stability and theoretical measure approximation.

Main Theoretical Results

Existence and Uniqueness of Numerical Invariant Measure

The analysis leverages moment bounds and contractivity estimates in Wasserstein-$b:\mathbb{R}^n \rightarrow \mathbb{R}^n$8 metrics to establish:

• Existence: The numerical process generated by the stochastic theta method possesses a tight sequence of probability measures, for which a weakly convergent subsequence attains an invariant measure.
• Uniqueness: An explicit exponential decay in Wasserstein-$b:\mathbb{R}^n \rightarrow \mathbb{R}^n$9 distance between trajectories initialized from distinct states ($g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$0), under the dissipativity assumption, guarantees uniqueness of the invariant measure.

The key innovation is the demonstration that, even for $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$1 with super-linear growth, the implicit structure of the stochastic theta method (for $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$2) yields sufficient stability and contractivity to control the explosion risk and ensure ergodicity in the numerical process.

Convergence Rate to the Exact Invariant Measure

By invoking finely tuned strong convergence rates for the stochastic theta method under monotonicity and local polynomial Lipschitz conditions [WangWuDong2020], the paper proves:

$g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$3

where $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$4 is the numerical invariant measure, $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$5 the true SDE invariant measure, and $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$6 the $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$7-Wasserstein distance. This quantifies the effectiveness of the approximation: as $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$8 decreases, the numerical invariant measure converges rapidly to its theoretical counterpart.

Generalization to Implicit Methods

Since the backward Euler-Maruyama method is encompassed as $g:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times d}$9, the results subsume earlier findings for backward EM [LiuMaoWu2023], but extend to the broader class of stochastic theta methods for cases where the diffusion coefficient $|x|^q$0 is also super-linear.

Numerical Experimentation

Two illustrative test problems validate the theory:

1. Scalar mean-reverting SDE with cubic nonlinearity and quadratic diffusion: Empirical histograms and Kolmogorov-Smirnov statistics verify stabilization and convergence to an invariant measure, independent of initialization.
2. 2D SDE with cubic drift and quadratic diffusion: Moment bounds and density evolution confirm the preservation of ergodicity and absence of divergent paths, demonstrating method utility in multi-dimensional, strongly nonlinear settings.

Implications and Future Directions

The work demonstrates that the stochastic theta method, given sufficient implicitness ($|x|^q$1), can approximate invariant measures of SDEs with coefficients beyond classical global Lipschitz conditions, facilitating robust simulation of systems exhibiting strong nonlinearities — including those relevant to population dynamics, generative diffusion models, and regime-switching processes.

Practical implications include:

• Robustness of numerical ergodic approximation: The method remains stable for wide classes of SDEs inherently difficult for explicit schemes due to super-linear growth.
• Extension to mixture sampling and state-dependent switching: The theoretical framework paves the way for ergodic approximation in complex mixture and regime-switching SDEs, enabling new sampling strategies for multimodal distributions relevant to machine learning [Tretyakov2025].

Theoretical future directions are suggested:

• Relaxing the coercivity imposed via second moments, potentially using $|x|^q$2th moments for small $|x|^q$3, may further broaden applicability [LiMaYangYuan2018].
• Systematic study of stochastic theta method adaptations for SDEs with state-dependent switching to enhance mixture model sampling and modeling of complex stochastic phenomena.

Conclusion

This paper rigorously establishes existence, uniqueness, and convergence rates for the numerical invariant measure generated by the stochastic theta method for SDEs with super-linearly growing drift and diffusion coefficients (2606.19886). The framework generalizes prior results, validates efficacy through numerical examples, and outlines substantive avenues for both methodological improvement and practical implementation in stochastic simulation and sampling.