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Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients
Martin Hutzenthaler, Arnulf Jentzen, Peter E. Kloeden
TL;DR
The paper addresses whether Euler's method converges at finite time for SDEs with superlinearly growing, non-globally Lipschitz coefficients. It analyzes the explicit Euler approximation and proves that, for a large class of such SDEs, both strong and numerically weak convergence fail, with relevant differences and moments diverging to infinity.
Problem
Finite-time strong or numerically weak convergence of Euler's approximation remained an open question for SDEs with non-globally Lipschitz, superlinearly growing coefficients.
Method
The paper studies the explicit Euler approximation for a large class of SDEs with superlinearly growing coefficients and applies its main divergence theorem to examples.
Results
For every p ∈ [1, ∞), both the strong Lp distance and the difference between p-th absolute moments diverge to infinity at finite time, so strong and numerically weak convergence fail.
Takeaways & Limitations
The Euler scheme does not approximate the exact solution in the strong or numerically weak sense for the considered class of SDEs at finite time.
Takeaways & Limitations
The stated coefficient condition requires either the drift or diffusion to grow at higher polynomial order than linearly while the other grows more slowly.
Abstract
from arXiv · showhide
The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz continuous drift and diffusion coefficient. Recent results extend this convergence to coefficients which grow at most linearly. For superlinearly growing coefficients finite-time convergence in the strong mean square sense remained an open question according to [Higham, Mao & Stuart (2002); Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40, no. 3, 1041-1063]. In this article we answer this question to the negative and prove for a large class of stochastic differential equations with non-globally Lipschitz continuous coefficients that Euler's approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense.
1. Introduction
The paper shows that, for a large class of SDEs with superlinearly growing coefficients, the explicit Euler approximation can diverge at finite time despite pathwise convergence results and established convergence under weaker growth conditions.
- Contribution: Finite-time strong Lp-divergence occurs for the Euler approximation for every p ∈ [1, ∞) in the considered class of SDEs.The result applies to the distance between the Euler approximation and the exact solution.
- Contribution: Finite-time numerically weak convergence also fails for the same class of SDEs.The paper distinguishes numerically weak convergence from stochastic weak convergence.
- Results: The absolute moments of the Euler approximation at a finite time point diverge to infinity for all p ∈ [1, ∞).Consequently, the moment-bound assumption used in prior strong mean square convergence results is not satisfied for the introductory SDE.
- Scope: The divergence is established for a large class of SDEs with superlinearly growing coefficients, rather than only for the introductory example.The estimates are also described as adaptable to other numerical schemes such as Milstein's method.
- Scope: Noise is essential to the reported divergence: in the deterministic case, Euler's scheme converges and the stated divergence results fail.This contrasts the stochastic finite-time behavior with the deterministic case.
- Related work: The paper contrasts its finite-time moment divergence with prior infinite-time divergence results, where the discretization step remains fixed and pathwise divergence occurs with positive probability.On finite intervals, pathwise convergence can still hold, so the moment divergence is not attributed to pathwise behavior.
2. Main result and examples
The main theorem shows that explicit Euler approximations can diverge at finite times for a broad class of SDEs with superlinearly growing coefficients, despite pathwise convergence results. Several stochastic models satisfy the theorem’s assumptions and therefore exhibit this strong and weak divergence.
- Setting: The Euler scheme is defined on a fixed finite interval for an SDE driven by Brownian motion, with measurable drift and diffusion coefficients.The exact solution is assumed to exist, and the approximation is given at discrete time points nT/N.
- Main theorem: If at least one coefficient grows superlinearly under the theorem’s conditions, Euler’s approximation diverges in strong Lp and numerically weak senses.The theorem provides events with exponentially small probability on which approximations grow double-exponentially, causing unbounded moments.
- Main theorem: The divergence mechanism requires noise and does not extend to the deterministic Euler scheme.The diffusion must be nonzero at the starting point, while deterministic Euler approximations converge in the discussed setting.
- Assumptions: The theorem applies when one coefficient has higher polynomial growth than linearly while the other grows more slowly.The assumptions allow either drift-dominated or diffusion-dominated superlinear growth, with the dominant exponent exceeding one.
3. Simulations
The simulations illustrate finite-time divergence of Euler’s method using stochastic differential equations with explicit solutions and superlinearly growing coefficients. The reported moments can grow double-exponentially or become numerically undefined.
- Simulation setup: The simulations use stochastic differential equations with explicit solutions to compare Euler approximations against exact behavior.The section presents two numerical simulations illustrating the divergence result.
- Simulation setup: For the Ginzburg–Landau example, Monte Carlo simulations compare the second moments of the exact solution and Euler approximation across five noise levels.Each noise level includes ten Euler simulation runs with N = 10^3 time steps.
- Ginzburg–Landau simulation: At noise level σ̄ = 7, the Euler simulation produces NaN, while simulations for σ̄ ∈ {2, 4, 5} remain finite.The reported NaN results from an Inf − Inf arithmetic operation.
- Ginzburg–Landau simulation: The table values are either within distance two of the true value or NaN, reflecting rapid growth once the deterministic system begins to increase.Similar behavior occurs for other exponents greater than two.
- Second simulation: For the second simulation, the first absolute moment over t ∈ [0, 10] grows close to double-exponentially on a logarithmic y-axis.The Euler scheme is simulated for N ∈ {1, …, 50} with 10^4 Monte Carlo runs.
4. Proofs
The proofs establish the divergence theorem through probabilistic estimates and induction on the Euler time steps. The argument concludes by combining the derived inequalities to complete the theorem.
- Supporting lemma: An auxiliary lemma for a standard normally distributed random variable supplies a key probabilistic estimate.The lemma is introduced before the theorem proof proceeds by induction.
- Proof of Theorem 1: The proof begins from positive probability that the diffusion coefficient at the initial random variable is nonzero.This yields a constant K > 1 used in the subsequent estimates.
- Proof of Theorem 1: The proof establishes the relevant inequality for every Euler step by induction, beginning with n = 1 and handling the step n → n + 1.The induction uses the growth condition and an earlier inequality.
- Proof of Theorem 1: Definitions of the auxiliary events and variables are combined with the growth condition to obtain the required bounds.The argument applies these bounds across all N ∈ N.
- Proof of Theorem 1: Combining the established inequalities completes the proof of Theorem 1.The final step explicitly combines inequalities (4.14) and (4.16).