Randomized-Tamed Milstein Scheme

• The paper introduces a randomized-tamed Milstein scheme that attains strong L^p convergence of order one under relaxed conditions, combining drift randomization with a taming strategy.
• Drift randomization compensates for limited Hölder temporal continuity, while the taming mechanism controls superlinear growth, ensuring stability and boundedness.
• The method's efficacy is demonstrated through numerical examples, restoring full order one convergence where classical methods typically plateau.

The randomized-tamed Milstein scheme is a numerical method for strong approximation of stochastic differential equations (SDEs) with drift coefficients exhibiting superlinear growth in the state variable and limited temporal regularity. This approach combines a taming mechanism—controlling state-dependent blowup of the drift—with a drift randomization strategy to compensate for merely Hölder temporal continuity. The method achieves an optimal strong $\mathscr{L}^p$-convergence rate of order one under suitably relaxed assumptions, overcoming the barriers posed by superlinear drifts and low regularity (Biswas, 14 Jan 2026).

1. Problem Framework and Structural Assumptions

Consider the SDE on $[0,T]$: $$\mathrm{d}X_t = a(t, X_t)\,\mathrm{d}t + b(t, X_t)\,\mathrm{d}W_t,\quad X_0 \in \mathbb{R}^d,$$ where $a$ may be superlinear in $x$ and only $\beta$-Hölder in $t$ for $\beta \in (0,1]$, while $b$ is globally Lipschitz in $x$ with at most linear growth. The underlying probability space $$(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, \mathbb{P})$$ supports an $m$-dimensional Brownian motion $W$.

The following technical conditions ensure uniqueness and finiteness of moments of the true solution $X_t$:

• Initial moment: $\mathbb{E}|X_0|^q < \infty$ for $q \ge 2$.
• Monotonicity-coercivity: For all $t \in [0,T]$, $x, y \in \mathbb{R}^d$,

$$\langle x-y,\, a(t,x) - a(t,y)\rangle \vee \|b(t,x) - b(t,y)\|^2 \le L |x-y|^2,\quad \|b(t,0)\| \le L$$

and $\langle x, a(t,x)\rangle \le L(1 + |x|^2)$.

• Polynomial Lipschitz in $x$ (superlinear drift): For $\xi\ge 0$,

$$|a(t,x) - a(t,y)| \le L(1 + |x| + |y|)^\xi |x-y|,\quad |a(t,0)| \le L.$$

• $\beta$-Hölder temporal regularity: For all $s, t \in [0,T]$, $x\in\mathbb{R}^d$,

$$|a(s,x) - a(t,x)| \le L|s-t|^\beta(1 + |x|^{\xi+1}),\quad \|b(s,x)-b(t,x)\| \le L|s-t|^\beta(1+|x|).$$

• Lipschitz diffusion: For all $t$, $x, y \in \mathbb{R}^d$,

$$\|b(t,x) - b(t,y)\| \le L|x-y|,\quad \|b(t,x)\| \le L(1+|x|).$$

These regularity conditions generalize classical settings significantly, in particular allowing for $a$ with superlinear state dependence and mere Hölder temporal continuity—a context where standard Euler and Milstein methods can diverge or lose strong order.

2. Time Discretization and Drift Randomization

Uniform time discretization at grid points $t_n = n h$ for $n = 0, ..., N$, $h = T/N$, is combined with stepwise drift evaluation at randomized times. In each interval $[t_n, t_{n+1})$:

• An independent $\tau_n \sim \text{Uniform}(0,1)$ is sampled, yielding random evaluation points $t_{n,\tau_n} = t_n + \tau_n h$.
• For any $s \in [t_n, t_{n+1})$, set $\eta(s) = t_n$, $\tau(s) = t_{n,\tau_n}$.

Randomization of the drift evaluation time addresses bias introduced by low temporal regularity in $a$. This device is crucial for restoring the full strong order when $a$ is merely Hölder—a regime where classical deterministic time-stepping would typically stall at order $\beta$ (Biswas, 14 Jan 2026).

3. Taming Mechanism

To control potential blowup from the superlinear drift, the method replaces $a(t,x)$ with a tamed version: $a_h(t, x) = \frac{a(t, x)}{1 + h |x|^{2\xi}}$ where $\xi$ is the superlinear exponent from the structural assumption on $a$. This yields

$$|a_h(t,x)| \le L,\quad \langle x, a_h(t,x) \rangle \le C(1 + |x|^2),$$

ensuring uniform boundedness of the drift while pointwise consistency is preserved: $a_h(t,x) \to a(t,x)$ as $h \to 0$. The taming bias $a(\tau_n,X_n)-a_h(\tau_n,X_n)$ is $O(h)$ and is explicitly controlled in the error analysis.

4. Scheme Definition and Algorithm

The one-step randomized-tamed Milstein update from $(t_n, X_n)$ to $X_{n+1}$ is given by: $$\begin{aligned} X_{n+1} =\;& X_n + h\,a_h(t_{n,\tau_n}, X_n) + b(t_n, X_n)\,\Delta W_n \ &+ \frac{1}{2} \sum_{j=1}^m L^{(j)}b(t_n, X_n)\left((\Delta W_n^j)^2 - h\right) \end{aligned}$$ where

$$L^{(j)}b^{(i)}(t,x) = \sum_{k=1}^d b_{k j}(t,x)\,\partial_{x_k} b_{i j}(t,x)$$

and $\Delta W_n = W_{t_{n+1}} - W_{t_n}$.

Key elements:

• Drift $a$ is evaluated at the random time $t_{n,\tau_n}$, with taming applied.
• Diffusion and Milstein corrections use the standard time grid value $(t_n, X_n)$.
• The update is explicit and computationally feasible.

An equivalent continuous-time formulation is provided but, for implementation, the discrete one-step scheme is used. Per time step, complexity is $O(d m)$ for evaluations of $b$ and its Jacobian, plus one extra uniform random variate $\tau_n$ per step.

Component	Evaluation Time	Modification
Drift term $a$	$t_{n,\tau_n}$	Tamed: $a_h$ at random time
Diffusion $b$	$t_n$	Standard Milstein correction at grid
Milstein term	$t_n$	Uses Jacobian and cross-terms

The randomization applies solely to the drift term; the Milstein correction ensures strong order one when $b$ is sufficiently smooth in $x$.

5. Convergence Theory

The central result is strong $\mathscr{L}^p$-convergence of order one under conditions: $$\max_{0 \le n \le N} \|X_{t_n} - X_n\|_{L^p} \le C h,$$ for all $p \ge 2$ and $2p(\xi+2) \le q$ (Biswas, 14 Jan 2026). The analysis proceeds by:

1. Introducing an auxiliary process $Z_t^h$ matching the SDE dynamic but with randomized drift,
2. Proving $\|X_t - Z_t^h\|_{L^p}\le Ch$ via the $\beta$-Hölder continuity of $a$ and a discrete martingale argument,
3. Showing $\|Z_t^h - X^h_t\|_{L^p}\le Ch$ by Itô’s formula, estimating both the taming error and the Milstein discretization error,
4. Employing a Grönwall-type argument to conclude the global order one result.

Critical intermediate estimates include uniform moment bounds, local error bounds per step, and tight control of the bias introduced by taming.

A notable implication is that, in contrast to classical tamed Milstein methods which achieve only order $\approx \beta$ when $a$ is $\beta$-Hölder in $t$, randomization of the drift evaluation recovers the full strong order $1$ rate.

6. Implementation Procedure

The practical implementation follows these steps for $n = 0, \dots, N-1$:

1. Sample $\tau_n \sim \mathrm{Uniform}(0,1)$.
2. Set $t_{n,\tau_n} = t_n + \tau_n h$.
3. Compute $$F = a_h(t_{n,\tau_n}, X_n) = a(t_{n,\tau_n}, X_n) / [1 + h |X_n|^{2\xi}]$$.
4. Sample $\Delta W_n \sim \mathcal{N}(0, h I_m)$.
5. Evaluate Milstein correction $$M = \sum_{j=1}^m L^{(j)}b(t_n, X_n)[(\Delta W_n^j)^2 - h]/2$$.
6. Update $X_{n+1} = X_n + hF + b(t_n, X_n)\Delta W_n + M$.

The method is computationally straightforward, requiring only one additional randomness source per time step relative to the standard Milstein method.

7. Numerical Illustration and Practical Behavior

Applied to the FitzHugh–Nagumo system—a coupled SDE with superlinear drift in $V$—the randomized-tamed Milstein scheme demonstrates near-optimal strong $L^2$ convergence close to order one. Specifically,

• For drift coefficients that are only $\beta$-Hölder in time, classical tamed Milstein methods plateau at order $\approx\beta$;
• The introduction of drift randomization restores the expected order one performance (Biswas, 14 Jan 2026).

This suggests that the randomized-tamed Milstein scheme effectively overcomes bias due to temporal irregularity, broadening the applicability of strong Milstein-type methods to a larger class of SDEs with irregular drift behavior.