[Paper Review] Weak convergence of Euler-Maruyama's approximation for SDEs under integrability condition

Jinghai Shao|arXiv (Cornell University)|Aug 22, 2018

Stochastic processes and financial applications24 references3 citations

TL;DR

This paper establishes the weak convergence of the Euler-Maruyama scheme for SDEs with singular drifts under an integrability condition instead of the standard growth condition. By leveraging the dimension-free Harnack inequality and Girsanov's theorem, it proves convergence with rate $ \delta^{1/2 \wedge \alpha} $, even for degenerate diffusions, and applies the method to a stochastic damping Hamiltonian system.

ABSTRACT

This work establishes the weak convergence of Euler-Maruyama's approximation for stochastic differential equations (SDEs) with singular drifts under the integrability condition in lieu of the widely used growth condition. This method is based on a skillful application of the dimension-free Harnack inequality. Moreover, when the drifts satisfy certain regularity conditions, the convergence rate is estimated. This method is also applicable when the diffusion coefficients are degenerate. A stochastic damping Hamiltonian system is studied as an illustrative example.

Motivation & Objective

• To establish weak convergence of the Euler-Maruyama approximation for SDEs with singular drifts when the classical growth condition fails.
• To replace the standard growth or Lyapunov conditions with a weaker integrability condition on the drift.
• To extend the convergence analysis to degenerate diffusion coefficients.
• To derive convergence rates under additional regularity assumptions on the drift.
• To demonstrate the method on a stochastic damping Hamiltonian system as a concrete example.

Proposed method

• Utilizes the dimension-free Harnack inequality to handle SDEs with singular drifts not satisfying standard growth conditions.
• Applies Girsanov's theorem to re-express the SDEs under equivalent probability measures, enabling the use of Harnack inequalities.
• Constructs auxiliary processes to decouple the drift and diffusion components for analysis.
• Employs time-discretized approximations via the Euler-Maruyama scheme and compares the law of the exact and approximate solutions.
• Establishes Novikov’s condition for Girsanov transformations using integrability and regularity assumptions.
• Derives convergence rates by bounding the difference in expectations of functionals via moment estimates and stochastic integrals.

Experimental results

Research questions

• RQ1 Can the Euler-Maruyama scheme converge weakly for SDEs with singular drifts that violate the standard growth condition?
• RQ2 Does the dimension-free Harnack inequality enable weak convergence analysis under weaker integrability conditions than growth or Lyapunov conditions?
• RQ3 Can the method be extended to degenerate diffusion coefficients where the diffusion matrix is not uniformly non-degenerate?
• RQ4 What is the convergence rate of the Euler-Maruyama scheme under additional regularity assumptions on the drift?
• RQ5 How does the method perform in concrete models such as stochastic damping Hamiltonian systems?

Key findings

• The Euler-Maruyama scheme converges weakly for SDEs with singular drifts under an integrability condition, even when the drift does not satisfy any growth or Lyapunov condition.
• The convergence rate is $ \delta^{1/2 \wedge \alpha} $, where $ \alpha $ is a regularity parameter of the drift, under additional smoothness assumptions.
• The method is applicable to degenerate SDEs, as demonstrated by the analysis of a stochastic damping Hamiltonian system.
• The dimension-free Harnack inequality is instrumental in verifying Novikov’s condition and controlling the Radon-Nikodym derivatives in Girsanov transformations.
• The convergence result holds for a class of drifts that are not in $ \mathbb{L}_p^q $, such as the example $ b(x) = \left( \sum_{n=1}^\infty \log\left(1 + \frac{1}{|x - n|^2}\right) \right)^{1/2} - x $.
• Moment estimates and stochastic integral bounds lead to the final convergence rate via the difference in expectations of functionals of the solution and its approximation.