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A Randomized Milstein Scheme for SDEs with Superlinear Drift Coefficient

Sani Biswas

TL;DR

The paper addresses numerical approximation of SDEs with superlinear drift and time-irregular coefficients by introducing a randomized-tamed Milstein scheme that combines drift randomization with a taming mechanism. The authors prove a strong $\mathscr{L}^p$-convergence rate of $\min\{\beta+\tfrac{1}{2},1\}$ under $\beta\in(0,1]$ and appropriate moment conditions, aided by an auxiliary process that connects the SDE and the numerical scheme. Key technical contributions include new estimates for randomized and taming terms and a novel two-step analysis that yields optimal rates, with a concrete taming example and auxiliary results underpinning stability. Numerical experiments on the FitzHugh–Nagumo system corroborate the theoretical rate, demonstrating practical applicability to time-inhomogeneous SDEs in neuroscience and related fields.

Abstract

This work presents a randomized-tamed Milstein scheme for stochastic differential equations whose drift coefficient exhibits superlinear growth in the state variable and limited temporal regularity, quantified by $β$-Hölder continuity with $β\in (0,1]$. The scheme combines a taming mechanism to control the superlinear state dependence with a drift randomization strategy designed to address the challenges posed by low temporal regularity. Under suitable assumptions on temporal smoothness, the scheme achieves an optimal strong $\mathscr{L}^p$-convergence rate of order one.

A Randomized Milstein Scheme for SDEs with Superlinear Drift Coefficient

TL;DR

The paper addresses numerical approximation of SDEs with superlinear drift and time-irregular coefficients by introducing a randomized-tamed Milstein scheme that combines drift randomization with a taming mechanism. The authors prove a strong -convergence rate of under and appropriate moment conditions, aided by an auxiliary process that connects the SDE and the numerical scheme. Key technical contributions include new estimates for randomized and taming terms and a novel two-step analysis that yields optimal rates, with a concrete taming example and auxiliary results underpinning stability. Numerical experiments on the FitzHugh–Nagumo system corroborate the theoretical rate, demonstrating practical applicability to time-inhomogeneous SDEs in neuroscience and related fields.

Abstract

This work presents a randomized-tamed Milstein scheme for stochastic differential equations whose drift coefficient exhibits superlinear growth in the state variable and limited temporal regularity, quantified by -Hölder continuity with . The scheme combines a taming mechanism to control the superlinear state dependence with a drift randomization strategy designed to address the challenges posed by low temporal regularity. Under suitable assumptions on temporal smoothness, the scheme achieves an optimal strong -convergence rate of order one.
Paper Structure (7 sections, 13 theorems, 64 equations, 1 figure)

This paper contains 7 sections, 13 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Result
  3. Randomized-Tamed Milstein Scheme and Main Result
  4. An Illustrative Example of Drift Taming
  5. Moment bound of the randomized-tamed Milstein scheme \ref{['eq:scm']}
  6. Proof of the Main Result
  7. Numerical Experiment

Key Result

Proposition 2.1

Suppose that Hypotheses asum:monotonocity to asum:continuity hold. Then the SDE eq:sde admits a unique strong solution. Moreover, if Assumption asum:ic also holds, there exists a constant $C > 0$, such that $\sup_{t \in [0,T]} \mathbb E|x_t|^q \leq C.$

Figures (1)

  • Figure 1: $\mathscr{L}^p$-error of the scheme \ref{['eq:tamed-milstein']} for the SDE \ref{['eq:sde-system']}.

Theorems & Definitions (27)

  • Proposition 2.1
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 3.1
  • ...and 17 more