A Randomized Milstein Scheme for SDEs with Superlinear Drift Coefficient

Technical Summary: A Randomized Tamed Milstein Scheme for SDEs with Superlinear Drift

Problem Statement
This work addresses the numerical approximation of stochastic differential equations (SDEs) of the form:
$$x_t = x_0 + \int_0^t \mu(s, x_s) ds + \int_0^t \rho(s, x_s) dw_s$$
where the drift coefficient $\mu$ exhibits superlinear growth in the state variable and possesses low temporal regularity, specifically quantified by $\beta$-Hölder continuity with $\beta \in (0, 1]$.

Classical explicit numerical schemes (such as the Euler-Maruyama or standard Milstein methods) fail in this setting. As demonstrated in prior literature, explicit schemes applied to SDEs with superlinearly growing coefficients often suffer from finite-time blow-up due to uncontrolled moment growth. While implicit methods offer stability, they incur significant computational overhead, particularly in high dimensions. Furthermore, standard explicit schemes struggle with time-irregular drifts, often failing to achieve optimal convergence rates.

Methodology
The authors propose a randomized-tamed Milstein scheme that integrates two distinct strategies to overcome the dual challenges of superlinear growth and low time regularity:

1. Taming Mechanism: To control the superlinear growth of the drift, the scheme employs a "taming" function. Specifically, the drift term is modified as:
   $$(\bar{\mu})^n_{tm}(s, x) := \frac{\mu(s, x)}{1 + n^{-1}|x|^{2\xi}}$$
   This modification ensures that the drift remains bounded relative to the discretization parameter $n$, preventing moment explosion while preserving the explicit nature of the scheme.

2. Drift Randomization: To address the low temporal regularity of the drift, the scheme utilizes a randomization strategy. Instead of evaluating the drift at the deterministic grid points $t_j$, the scheme evaluates the drift at random time points $\tau^n_s = t_{j-1} + \delta t \cdot \tau_{j-1}$, where $\tau_{j-1}$ are independent uniform random variables on $[0, 1]$. This randomization leverages the integral properties of the drift to recover higher-order convergence rates despite the lack of smoothness in time.

The resulting scheme (Equation 3) approximates the solution $x_t$ via:
$$x^n_t = x_0 + \int_0^t (\bar{\mu})^n_{tm}(\tau^n_s, x^n_{\eta^n_s}) ds + \int_0^t \Gamma(\eta^n_s, s, x^n_{\eta^n_s}) dw_s$$
where $\Gamma$ represents the Milstein correction term involving the derivative of the diffusion coefficient $\rho$.

Key Contributions and Analytical Framework
The primary contribution is the rigorous proof of strong $L^p$-convergence for this combined scheme. The analysis introduces several novel elements:

• Auxiliary Process: The authors define an auxiliary process $z^n_t$ (Equation 7) governed by the original drift evaluated at random times but without the taming or Milstein correction. This process serves as a crucial bridge between the true solution $x_t$ and the numerical scheme $x^n_t$.
• Decomposition of Error: The convergence proof relies on splitting the error $|x_t - x^n_t|$ into two components: $|x_t - z^n_t|$ (error due to randomization) and $|z^n_t - x^n_t|$ (error due to taming and discretization).
• New Estimates: The paper derives essential estimates (Lemmas 3.1 and 3.5) that handle the interaction between the taming mechanism and the randomization. These estimates are necessary because existing convergence theories for either randomized schemes or tamed schemes individually are insufficient for this combined setting.

Main Results
Under suitable hypotheses regarding the moments of the initial condition, the growth of the coefficients, and the Hölder continuity of the drift and diffusion, the paper establishes the following convergence rate (Theorem 2.1):

$$\sup_{t \in [0, T]} \mathbb{E}|x_t - x^n_t|^p \leq C n^{-p \min\{\beta + \frac{1}{2}, 1\}}$$

• Optimality: When the drift is sufficiently regular ($\beta \geq 1/2$), the scheme achieves an optimal strong convergence rate of order 1.
• Comparison: This rate surpasses the convergence rate of the classical tamed Milstein scheme, which is limited to order $\beta$ in the presence of time-irregular drift.
• Applicability: The results hold for SDEs with superlinear drift and linear diffusion. The paper notes that while the diffusion coefficient in the referenced taming literature [17] could be superlinear, the current randomized-tamed Milstein scheme is restricted to the linear diffusion case.

Significance and Scope
The paper claims to present the first rigorous study of a randomized version of the tamed Milstein scheme for SDEs with superlinear coefficients. This work extends the novelty of randomized numerical methods to the context of superlinear growth, a setting previously unaddressed by such techniques.

The analytical framework developed is intended to serve as a foundation for future investigations into:

• Higher-order randomized-tamed schemes for Lévy-driven SDEs with superlinear growth.
• McKean–Vlasov type SDEs.

Numerical Validation
The theoretical findings are validated through numerical experiments on the stochastic FitzHugh–Nagumo system, a model used in neuroscience. The system features a cubic nonlinearity in the drift (superlinear growth) and time-dependent external inputs. The numerical results confirm the theoretical prediction, exhibiting a convergence rate of order 1.0, consistent with the case where the drift possesses sufficient temporal regularity.

Limitations
The authors explicitly state that the current scheme covers only the linear diffusion case. While the taming mechanism in prior work [17] accommodated superlinear diffusion, the combination of randomization and taming in this specific Milstein framework is currently limited to linear diffusion coefficients. Additionally, the analysis focuses on jump-free SDEs, excluding Lévy noise, which is noted as a direction for future work.