Sine-Milstein Method for Taming SDEs

• Sine-Milstein Method is a numerical scheme for approximating stochastic differential equations with superlinear drift and diffusion by using sine-based transformations to tame coefficient growth.
• It achieves robust order-one strong convergence and moment stability by replacing unbounded coefficient updates with bounded sine-modulated increments.
• The method improves numerical stability over classical Milstein and Euler-Maruyama schemes, making it effective for high-dimensional systems and measure-dependent SDEs.

The Sine-Milstein Method is a member of a unified class of Milstein-type numerical schemes designed for the strong approximation of stochastic differential equations (SDEs) and McKean-Vlasov SDEs (MV-SDEs) when drift and diffusion coefficients exhibit superlinear growth. In contrast to classical Milstein and Euler-Maruyama methods, which may diverge or yield unbounded numerical solutions in such settings, the Sine-Milstein methodology applies nonlinear "taming" mappings—specifically, sine-based transformations—to the SDE coefficients and their derivatives. This approach ensures bounded coefficient increments and preserves the essential higher-order corrections required for Milstein-type schemes, thus attaining robust, order-one strong convergence under mild regularity assumptions.

1. Motivation for Milstein-Type Schemes with Nonlinear Coefficient Growth

Classical explicit schemes in SDE simulation, including Euler-Maruyama and Milstein, rely on globally Lipschitz continuity of the drift and diffusion coefficients. However, many MV-SDEs arising from collective dynamics, physical systems, or nonlinear models have coefficients—such as $f(x)$ or $g(x)$—growing faster than linearly, which compromises numerical stability and may lead to particle corruption when simulating large interacting particle systems. The Sine-Milstein method addresses this by modifying the numerical update so that the dominant nonlinear increments are tamed without sacrificing the strong convergence order inherent in Milstein schemes (Zhu et al., 19 Oct 2025).

2. Sine-Based Taming of Coefficients in Milstein-Type Updates

In the Sine-Milstein framework, each coefficient function $\mathcal{F}$ occurring in drift ($f$), diffusion ($g_j$), or their derivatives within the Milstein correction term is replaced by a sine-modulated surrogate: $$\Gamma_l(\mathcal{F}, \Delta t) = \frac{1}{\Delta t} \sin(\Delta t \cdot \mathcal{F}) \qquad l = 1,2,3,4$$ The discrete-time update for each particle $i$ in an interacting particle system takes the form: $$Y_{t_{k+1}}^{(i,N)} = Y_{t_k}^{(i,N)} + \Gamma_1(f(Y_{t_k}^{(i,N)}, \rho_{t_k}^{(Y,N)}), \Delta t) \cdot \Delta t + \sum_{j=1}^m \Gamma_2(g_j(Y_{t_k}^{(i,N)}, \rho_{t_k}^{(Y,N)}), \Delta t) \cdot \Delta W_{t_k}^{(i,j)} + \text{Milstein correction terms}$$ where $\rho_{t_k}^{(Y,N)}$ is the empirical measure over the particle system and the higher-order Milstein terms are treated with analogous sine-based taming (Zhu et al., 19 Oct 2025).

The sine mapping ensures that for small arguments (i.e., small time steps or moderate coefficient values), the scheme closely approximates the classical update ($\sin(\Delta t f) \approx \Delta t f$), thus preserving accuracy. For large coefficients, the boundedness of $\sin(\cdot)$ constrains increments and prevents numerical explosion.

3. Theoretical Error Bounds and Strong Convergence Properties

Analysis in (Zhu et al., 19 Oct 2025) establishes the following strong error bound for the Sine-Milstein scheme in the interacting particle system setting: $$\sup_{t \in [0,T]} \sup_{i} \mathbb{E}\left[ |X_t^{(i,N)} - Y_t^{(i,N)}|^p \right] \leq C (1 + \mathbb{E}[|Y_0|^{\bar{p}}]^\beta) \Delta t^p$$ for some exponent $\beta$ and suitable $p$ in a prescribed interval, and with $C$ depending on canonical system parameters. This order-one rate matches the classical Milstein method under globally Lipschitz coefficients, but remains valid even for superlinear drift or diffusion, due to the uniform boundedness induced by the sine mapping.

Moreover, the propagation of chaos result links the approximation error of the MV-SDE to the empirical measure error and the Milstein step size: $$\sup_{t \in [0,T]} \sup_{i} \mathbb{E}\left[ |X_t^{(i)} - Y_t^{(i,N)}|^2 \right] \leq C( \eta_d(N) + \Delta t^2 )$$ where $\eta_d(N)$ is the standard empirical law approximation rate (Zhu et al., 19 Oct 2025).

4. Relationship with Other Tamed Milstein-Type Methods

The general framework in (Zhu et al., 19 Oct 2025) encompasses various tamed, tanh-, and mixed-taming Milstein schemes. For example, the Tamed-Milstein employs

$$\Gamma_l(\mathcal{F}, \Delta t) = \frac{\mathcal{F}}{1 + \Delta t |\mathcal{F}|}$$

and the Tanh-Milstein uses

$$\Gamma_l(\mathcal{F}, \Delta t) = \frac{1}{\Delta t} \tanh(\Delta t \mathcal{F})$$

All share the property of being globally bounded/tamed in a time-step dependent manner; the sine mapping is particularly well-suited for oscillatory or periodic systems, and its strict uniform boundedness provides stability in the presence of heavy superlinear growth.

5. Numerical Results and Applications

Numerical experiments in (Zhu et al., 19 Oct 2025) demonstrate the stability and efficiency of the Sine-Milstein scheme on MV-SDEs with superlinear coefficients. In mean-field double-well models, the method consistently produces stable trajectories approaching the expected equilibrium regions (e.g., neighborhoods of $1$ and $-1$); divergence commonly observed in classical Milstein or Euler schemes is avoided. Root-mean-square error log-log plots confirm first-order strong convergence, independent of the severity of nonlinear growth, for a variety of parameters and particle numbers $N$. Similar results are observed in high-dimensional models, including flocking and neuronal systems.

6. Implementation Considerations and Extensions

The Sine-Milstein scheme is explicit, requiring only standard Milstein-type terms with sine-based coefficient taming. Since the sine mapping is smooth and preserves local consistency, it can be implemented directly in existing SDE solvers or particle system codes by substituting the tamed update. The approach generalizes readily to Milstein-type schemes for SDEs with measure-dependent coefficients and can be integrated into advanced frameworks, such as multi-level Monte Carlo, where strong convergence rate improvements are critical (Zhu et al., 19 Oct 2025). The method is applicable wherever high-order strong approximation and robust stability under superlinear growth are required—including nonlinear dynamics, collective phenomena, and high-dimensional stochastic models.

7. Limitations and Theoretical Scope

The Sine-Milstein method requires only mild differentiability assumptions on the coefficient functions (once differentiable suffices), avoids the need for continuous-time Itô calculus by relying on discrete-time expansion and binomial-type arguments, and provides uniform moment bounds under appropriate initial data. While optimal for strong convergence and moment stability, in practice, the benefits relative to other tamed Milstein variants are most pronounced for systems where the coefficients exhibit severe nonlinear growth or oscillation; models with solely mild nonlinearities may achieve similar performance with classical Milstein or other explicit schemes.

Summary Table: Milstein-Type Modification Operators

Scheme	Taming Operator $\Gamma(\mathcal{F},\Delta t)$	Boundedness Control
Classical Milstein	$\mathcal{F}$	None (may diverge for superlinear)
Tamed-Milstein	$\mathcal{F}/(1 + \Delta t |\mathcal{F}|)$	Weak, continuous monotonic
Tanh-Milstein	$(1/\Delta t)\tanh(\Delta t\mathcal{F})$	Strong, saturates at 1
Sine-Milstein	$(1/\Delta t)\sin(\Delta t\mathcal{F})$	Strong, oscillatory, uniformly 1

In conclusion, the Sine-Milstein method offers a theoretically justified, practically robust, and easily implementable solution to the simulation of nonlinear and measure-dependent SDEs with superlinear coefficients, ensuring order-one strong convergence and moment control through sine-based taming of all critical update terms (Zhu et al., 19 Oct 2025).