---
title: Randomized-Tamed Milstein Scheme
url: https://www.emergentmind.com/topics/randomized-tamed-milstein-scheme
type: topic
---

# Randomized-Tamed Milstein Scheme

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 [2601.09437].

## 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 $\beta$ [2601.09437].

## 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 $2p(\xi+2) \le q$ [2601.09437]. 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 [2601.09437].

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.

Source: https://www.emergentmind.com/topics/randomized-tamed-milstein-scheme