[Paper Review] The Euler scheme for stochastic differential equations with discontinuous drift coefficient: A numerical study of the convergence rate

Simone Göttlich, Kerstin Lux+1 more|arXiv (Cornell University)|May 12, 2017

Stochastic processes and financial applications37 references7 citations

TL;DR

This paper investigates the numerical convergence of the Euler scheme for stochastic differential equations (SDEs) with piecewise constant drift and constant diffusion, demonstrating that convergence rates depend critically on drift coefficient properties and initial conditions. It establishes that inward-pointing drifts yield stable, higher-order convergence due to ergodicity, while outward-pointing drifts show unreliable convergence estimates, and validates these findings via a rank-based stock market model.

ABSTRACT

The Euler scheme is one of the standard schemes to obtain numerical approximations of stochastic differential equations (SDEs). Its convergence properties are well-known in the case of globally Lipschitz continuous coefficients. However, in many situations, relevant systems do not show a smooth behavior, which results in SDE models with discontinuous drift coefficient. In this work, we will analyze the long time properties of the Euler scheme applied to SDEs with a piecewise constant drift and a constant diffusion coefficient and carry out intensive numerical tests for its convergence properties. We will emphasize on numerical convergence rates and analyze how they depend on properties of the drift coefficient and the initial value. We will also give theoretical interpretations of some of the arising phenomena. For application purposes, we will study a rank-based stock market model describing the evolution of the capital distribution within the market and provide theoretical as well as numerical results on the long time ranking behavior.

Motivation & Objective

• To analyze the numerical convergence behavior of the Euler scheme for SDEs with discontinuous (piecewise constant) drift coefficients.
• To investigate how convergence rates depend on drift coefficient properties such as direction (inward/outward pointing) and jump height.
• To examine the influence of initial values on convergence stability and accuracy.
• To validate theoretical findings using a rank-based stock market model, focusing on long-time ranking dynamics and occupation rates.
• To compare the Euler scheme with higher-order schemes in the context of irregular coefficients, assessing their numerical performance.

Proposed method

• The study employs the Euler scheme for time-discretized SDEs with piecewise constant drift and additive noise, defined by $ dX_t = \sum_{j=1}^s \alpha_j \mathbbm{1}_{B_j}(X_t) dt + \sigma dW_t $.
• Numerical convergence rates are estimated using a reference solution with fine time step $ \Delta = 2^{-14} $, comparing approximations at coarser steps.
• Ergodicity of the Euler scheme is analyzed using theory of Markov chains, particularly for inward-pointing drifts where long-term stability emerges.
• The rank-based stock market model is simulated using the Euler scheme to study long-time capital distribution and occupation rates.
• Monte Carlo simulations with $ M = 10^3 $ repetitions are used to estimate discrete occupation rates and compare them to analytical asymptotic values.
• Higher-order schemes (e.g., Milstein-type) are tested, but no improvement in convergence behavior is observed for discontinuous drifts.

Experimental results

Research questions

• RQ1 What is the empirical convergence rate of the Euler scheme for SDEs with piecewise constant drift and constant diffusion?
• RQ2 How do drift coefficient properties—particularly inward vs. outward pointing—impact the stability and accuracy of numerical convergence estimates?
• RQ3 To what extent does the initial value influence the observed convergence rate in the presence of discontinuous drift?
• RQ4 How well does the Euler scheme capture the long-time ranking behavior in a rank-based stock market model?
• RQ5 Do higher-order numerical schemes outperform the Euler scheme for SDEs with discontinuous drift coefficients?

Key findings

• For inward-pointing drift coefficients, the Euler scheme exhibits stable and higher-order numerical convergence rates, independent of initial conditions, due to the ergodicity of the underlying SDE and its approximation.
• Convergence rates for outward-pointing drifts are unreliable and fail to stabilize, with no clear asymptotic regime reached in numerical experiments.
• Discrete occupation rates in the rank-based stock market model converge to the analytical asymptotic value of $ 1/d = 1/3 $ as the time horizon $ T $ increases, with faster convergence for less varying initial capitalizations.
• With $ \Delta = 2^{-14} $, the discrete occupation rates are accurate to four digits, and only one-third of the 90 rates differ in the fifth digit when $ \Delta = 2^{-10} $, indicating diminishing returns from further step size refinement.
• Higher-order schemes such as the Milstein-type do not improve convergence behavior for discontinuous drifts, suggesting the Euler scheme remains a practical choice despite its simplicity.
• The sum of squared deviations from the analytical occupation rate drops from 0.2116 (initial $ Y(0) $) to 0.0001 (for $ T=1000 $), confirming convergence to the theoretical limit.