Sine-Euler Scheme: Taming Super-linear SDEs

• Sine-Euler scheme is an explicit Euler-type method designed for Lévy-driven McKean-Vlasov SDEs with super-linear coefficients, using a sine transformation.
• The method applies the transformation Γ(z,Δt)=Δt⁻¹sin(Δtz) to tame super-linear growth and stabilize drift, diffusion, and jump terms without coercivity conditions.
• It achieves uniform moment bounds and near 1/2 mean-square convergence in interacting particle approximations, supported by rigorous error analyses and numerical tests.

Searching arXiv for the cited paper and closely related work to ground the article. arxiv_search(query="(Zhu et al., 11 Sep 2025)", max_results=5, sort_by="submittedDate") arxiv_search(query="Euler-type methods for Levy-driven McKean-Vlasov SDEs with super-linear coefficients mean-square error analysis", max_results=5, sort_by="submittedDate") The sine-Euler scheme is an Euler-type numerical method for interacting particle approximations of Lévy-driven McKean-Vlasov stochastic differential equations with super-linear coefficients. In the formulation analyzed in "Euler-type methods for Levy-driven McKean-Vlasov SDEs with super-linear coefficients: mean-square error analysis" (Zhu et al., 11 Sep 2025), it appears as a special case of a general transformed Euler framework in which the drift, diffusion, and jump coefficients are passed through the map $$\Gamma_\ell(z,\Delta t)=\Delta t^{-1}\sin(\Delta t z)$$, $\ell=1,2,3$. This transformation tames super-linear growth, yields uniform moment bounds for the numerical solution, and supports a mean-square convergence theory with rate arbitrarily close to $1/2$ for the associated interacting particle systems, without requiring the coercivity condition used in earlier approaches such as Assumption B-1 in Kumar et al. (Neelima et al., 2020).

1. Problem class and interacting-particle formulation

The scheme is studied for Lévy-driven McKean-Vlasov SDEs through their interacting-particle system

$$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$

with empirical measure

$$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$

This formulation couples each particle to the empirical law of the full system rather than to a fixed external distribution. The coefficients $f$, $g$, and $h$ may grow super-linearly in the state variable, which is precisely the regime in which direct explicit Euler discretizations become difficult to control. The analysis in (Zhu et al., 11 Sep 2025) is built around this interacting-particle representation rather than a decoupled finite-dimensional SDE, so both the discretization error and the mean-field approximation error enter the final estimates.

2. Definition of the sine-Euler update

For a uniform time step $\Delta t$ and grid $t_k=k\Delta t$, the sine-Euler approximation $\ell=1,2,3$0 is defined by choosing

$\ell=1,2,3$1

The one-step update for particle $\ell=1,2,3$2 is

$\ell=1,2,3$3

Equivalently,

$\ell=1,2,3$4

The numerical state therefore evolves by replacing each raw coefficient evaluation with its sine-transformed analogue. In the paper’s general framework, this puts the sine-Euler method alongside the tanh-Euler and tamed-Euler schemes as particular instances of transformed explicit Euler discretizations (Zhu et al., 11 Sep 2025).

3. Taming mechanism and its analytical role

The central property of the sine map is the bound

$\ell=1,2,3$5

Because $\ell=1,2,3$6 is globally bounded by $\ell=1,2,3$7 and also by $\ell=1,2,3$8, the transformation simultaneously preserves small-argument behavior and truncates large-argument growth. The paper states that this taming acts on $\ell=1,2,3$9, $1/2$0, and $1/2$1 and does so without requiring a coercivity condition (Zhu et al., 11 Sep 2025).

A key consequence is that the drift increment remains of order $1/2$2 even when $1/2$3 grows like $1/2$4. This is the mechanism by which the method stabilizes explicit time stepping in the super-linear regime. The same principle extends to the diffusion and jump terms through $1/2$5 and $1/2$6, so the scheme is not merely drift-tamed; it is coefficient-wise transformed across all three channels.

The analytical framework assumes Assumptions 2.1–2.8 on $1/2$7, $1/2$8, and $1/2$9, including coupled monotonicity, polynomial growth, and Hölder-in-time regularity, together with Assumptions 4.1–4.2 on the transform $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$0. The paper’s significance lies in showing that these ingredients suffice for the error analysis, whereas existing approaches cited there rely on a coercivity condition such as Assumption B-1 in Kumar et al. (Neelima et al., 2020).

4. Uniform moment bounds and strong convergence

The scheme admits discrete-time and continuous-time moment bounds. By an inductive argument in Lemma 4.4, for each $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$1 large enough there exist constants $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$2 such that for every integer $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$3 in a suitable range,

$$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$4

Lemma B.2 extends this to all $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$5: $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$6

These bounds are described as crucial for controlling increments and remainder terms in the error analysis. They also provide the regularity needed to compare the numerical interpolant to the exact interacting-particle system.

The main strong-error statement, Theorem 4.6, asserts that for any $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$7 there is a constant $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$8 independent of $$X^{i,N}(t)=X^i_0+\int_0^t f\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,ds+\int_0^t g\bigl(s,X^{i,N}(s),\rho_s^{X,N}\bigr)\,dW^i(s)+\int_0^t\!\!\int_{\mathcal E} h\bigl(s,X^{i,N}(s),\rho_s^{X,N},v\bigr)\,\tilde p_\varphi^i(dv,ds),$$9 and $$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$0 such that

$$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$1

The paper emphasizes that this yields a rate arbitrarily close to $$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$2 in the mean-square sense. Combined with the propagation-of-chaos estimate of Proposition 3.1, Corollary 4.7 gives

$$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$3

with similar rates if $$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$4 (Zhu et al., 11 Sep 2025).

5. Structure of the convergence proof

The proof strategy begins with uniform moment bounds for the discrete scheme. The argument exploits the fact that $$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$5 is bounded by $$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$6 while retaining a Lipschitz-like behavior for small arguments. The paper presents this as the step that removes the need for coercivity assumptions.

The next stage compares the continuous-time interpolant of the sine-Euler approximation with the true interacting-particle system through an Itô-formula for the difference process. The enhanced coupled-monotonicity condition, Assumption 4.5, yields a one-sided bound of the form

$$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$7

Remainder terms are then controlled using the sine-map approximation error,

$$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$8

Lemma B.3 gives the local error for each coefficient as being of order $$\rho_s^{X,N}=\frac1N\sum_{j=1}^N\delta_{X^{j,N}(s)}.$$9. A Gronwall argument then closes the estimate and produces the global rate $f$0 (Zhu et al., 11 Sep 2025). A plausible implication is that the sine transformation is not merely a bounded surrogate for the identity; it is calibrated so that the local consistency loss remains compatible with a near-$f$1 strong rate.

6. Numerical behavior and relation to adjacent schemes

Section 5 of the paper reports two representative numerical examples. In Example 5.1, a 3/2-volatility model on $f$2 with jump-intensity $f$3 and initial value $f$4 is simulated: $f$5 A reference solution uses time-step $f$6 and $f$7 i.i.d. samples. The sine-Euler method, together with tanh, tamed, and mixed variants, is run with $f$8 and $f$9 particles. On a log-log plot of MSE versus $g$0, all four methods show slopes nearly $g$1.

In Example 5.2, a double-well model on $g$2 with $g$3 and $g$4 is considered: $g$5 Using the same Monte Carlo protocol, with $g$6 and $g$7, the sine-Euler curve is again parallel to a reference line of slope $g$8 (Zhu et al., 11 Sep 2025).

These experiments are presented as validation of the theoretical rate and as evidence that the method performs comparably to the other transformed Euler variants included in the same general framework. A common misconception is that stabilization of super-linear McKean-Vlasov dynamics necessarily requires implicitness or coercivity-based arguments; in the setting studied here, the sine-Euler scheme provides an explicit counterexample to that presumption by combining bounded coefficient transforms with strong-error estimates.

7. Position within the transformed Euler framework

The sine-Euler scheme is one member of a broader class of Euler-type numerical schemes derived by incorporating projections or nonlinear transformations into the classical Euler method. In the paper’s taxonomy, the class includes the tanh-Euler, tamed-Euler, and sine-Euler schemes as special cases, all designed with the primary objective of establishing moment bounds for numerical solutions under super-linear coefficient growth (Zhu et al., 11 Sep 2025).

Within that class, the sine-Euler choice is distinguished by the elementary transformation

$g$9

Its appeal is structural rather than heuristic: the map is explicit, bounded, and directly compatible with the discrete-time error analysis developed for Lévy-driven McKean-Vlasov systems. The paper concludes that the scheme uses this elementary map to tame super-linear growth, provably enjoys uniform-in-time moment bounds without coercivity, converges in mean square with rate arbitrarily close to $h$0, and performs well in numerical tests. This suggests that the sine-Euler construction is best understood not as an isolated discretization trick, but as a concrete realization of a general principle for explicit approximation of mean-field jump-diffusion systems with super-linear nonlinearities.