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The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients
Xiaojie Wang, Siqing Gan
TL;DR
Classical explicit Euler schemes can fail to converge strongly for SDEs with superlinearly growing, globally one-sided Lipschitz drift coefficients. This paper introduces an explicit tamed Milstein method for commutative-noise SDEs, proves strong convergence of order one, and demonstrates computational efficiency over tamed Euler.
Problem
Classical explicit Euler schemes fail to converge strongly for SDEs with superlinearly growing, globally one-sided Lipschitz drift coefficients, motivating explicit strongly convergent alternatives.
Method
The paper introduces an explicit tamed Milstein scheme for SDEs with commutative noise and uses a bootstrap argument in its convergence analysis.
Results
Strong convergence order one is established under the stated regularity and Lipschitz assumptions, while an illustrative example confirms computational efficiency over tamed Euler.
Takeaways & Limitations
For commutative-noise SDEs, the tamed Milstein method provides a higher-order explicit alternative whose computational efficiency is demonstrated relative to tamed Euler.
Takeaways & Limitations
The approach is restricted to commutative noise and requires additional assumptions on coefficient regularity and Lipschitz continuity.
Abstract
from arXiv · showhide
For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the classical explicit Euler scheme fails to converge strongly to the exact solution. Recently, an explicit strongly convergent numerical scheme, called the tamed Euler method, is proposed in [Hutzenthaler, Jentzen & Kloeden, Ann. Appl. Probab., 22 (2012), pp. 1611-1641.] for such SDEs. Motivated by their work, we here introduce a tamed version of the Milstein scheme for SDEs with commutative noise. The proposed method is also explicit and easily implementable, but achieves higher strong convergence order than the tamed Euler method does. In recovering the strong convergence order one of the new method, new difficulties arise and kind of a bootstrap argument is developed to overcome them. Finally, an illustrative example confirms the computational efficiency of the tamed Milstein method compared to the tamed Euler method.
1 Introduction
The paper studies strong approximation of SDEs with non-globally Lipschitz drift, where explicit Euler can fail, and introduces a tamed Milstein method for commutative noise. The method is designed to retain explicit implementation while achieving strong order one under stated coefficient assumptions.
- Motivation: Explicit Euler fails to converge strongly for SDEs with superlinearly growing drift coefficients, whereas backward Euler is convergent but implicit.The tamed Euler method provides an explicit alternative with strong convergence order 0.5 under global one-sided Lipschitz and polynomial derivative conditions.
- Method: The proposed tamed Milstein method replaces the drift µ(Y_n) in the explicit Milstein scheme with a modified drift ˜µ(Y_n).This follows the taming strategy previously used for the tamed Euler method.
- Method: Commutative noise makes the Milstein method easily implementable without simulating iterated Itô integrals, improving its computational efficiency relative to Euler-type methods.The commutativity condition allows iterated-integral terms to be expressed using Wiener increments.
- Theory: The main result establishes strong convergence order one when the drift is globally one-sided Lipschitz with polynomially growing first and second derivatives, while diffusion-related coefficients are globally Lipschitz.The analysis obtains uniform boundedness of p-th moments and uses a bootstrap argument to control a sharper error estimate.
- Contribution: The paper presents the tamed Milstein method as the first successful recovery of strong order one for a Milstein-type method under non-globally Lipschitz conditions.The paper identifies a sharper estimate of term J as the key analytical difficulty.
- Evaluation: An illustrative example confirms strong order one and computational efficiency compared with the tamed Euler method.The example uses runtime and root-mean-square error comparisons to assess both methods.
2 Uniform boundedness of p-th moments
This section establishes uniform boundedness of p-th moments for the tamed Milstein numerical solution under the paper’s assumptions, using auxiliary events, dominating processes, martingale inequalities, and a bootstrap argument.
- Assumptions: Assumption 2.1 requires a globally one-sided Lipschitz drift, globally Lipschitz diffusion and Milstein coefficient, and polynomial growth of the drift derivative.The conditions are expressed through inequalities (2.1)–(2.4).
- Moment-boundedness strategy: The analysis introduces subevents Ω_n and dominating stochastic processes D_n to prove uniform boundedness of p-th moments.These constructions follow ideas from prior work and organize the moment estimates.
- Moment estimates: The proof controls the numerical increments on events where the Wiener increments are bounded, combining coefficient estimates with inequalities for inner products and norms.On Ω_n+1, the argument uses ∥ΔW_n∥≤1 together with Lipschitz and polynomial-growth bounds.
- Stochastic-term control: Martingale, submartingale, maximal, Hölder, Cauchy–Schwarz, and Burkholder–Davis–Gundy inequalities control the stochastic terms in the recursive estimates.The proof explicitly invokes discrete martingale properties and Doob’s maximal inequality.
- Bootstrap closure: A preliminary estimate contains N on its right-hand side, so a bootstrap argument is used before Gronwall’s lemma completes the moment bound.The paper identifies the N-dependence as insufficient and then combines subsequent estimates to finish the proof.
3 Strong convergence order of the tamed Milstein method
The paper proves strong convergence of the tamed Milstein method by analyzing continuous interpolations and Taylor remainders under additional smoothness assumptions. The proof combines moment bounds, martingale inequalities, remainder estimates, and Gronwall’s lemma.
- Additional assumptions: Additional convergence assumptions require twice continuously differentiable coefficients with polynomial-growth derivative bounds.These assumptions support the Taylor expansions and remainder estimates used in the convergence proof.
- Continuous interpolation: The analysis introduces a time-continuous interpolation that coincides with the discrete approximation at grid points and admits an integral representation.This representation enables direct comparison with the exact SDE solution.
- Convergence conclusion: Gronwall’s inequality establishes the desired result first for p≥4, while Hölder’s inequality extends the assertion to 1≤p<4.The final proof explicitly uses this two-stage argument.
- Error decomposition: Deterministic Taylor expansions and remainder terms decompose the drift and diffusion discrepancies needed to compare the numerical and exact solutions.The remainders are defined through integrals involving first and second derivatives.
- Error estimates: The convergence proof controls stochastic and remainder terms using coefficient assumptions, Hölder inequalities, Burkholder–Davis–Gundy inequalities, and discrete martingale estimates.The argument estimates the terms arising after applying Itô’s formula to the error representation.
4 An illustrative example
The illustrative experiment compares tamed Milstein and tamed Euler on a simple SDE using root-mean-square errors and runtime. The results show order-one error decay and a substantially lower runtime for tamed Milstein at the target precision.
- Runtime comparison: At precision ε=0.001, tamed Milstein reaches the target with N=2^10, whereas tamed Euler requires N=2^16.Figure 2 uses 1000 independent paths to approximate the expectation.
- Runtime comparison: At ε=0.001, tamed Milstein requires 8.1860 seconds compared with 147.9230 seconds for tamed Euler.For the tested commutative-noise SDE, the paper reports tamed Milstein as much faster.