[Paper Review] Euler-Maruyama Approximations for Stochastic McKean-Vlasov Equations with Non-Lipschitz Coefficients

Xiaojie Ding, Huijie Qiao|arXiv (Cornell University)|Mar 28, 2019

Stochastic processes and financial applications8 references4 citations

TL;DR

This paper establishes the existence and uniqueness of strong solutions for stochastic McKean-Vlasov equations with non-Lipschitz coefficients using Euler-Maruyama approximations. It proves optimal strong convergence rate of order $ O(2^{-n}) $ under linear growth and non-Lipschitz conditions, avoiding complex martingale solution constructions via direct approximation techniques.

ABSTRACT

In this paper we study a type of stochastic McKean-Vlasov equations with non-Lipschitz coefficients. Firstly, by an Euler-Maruyama approximation existence of its weak solutions is proved. And then we observe pathwise uniqueness of its weak solutions. Finally, it is shown that the Euler-Maruyama approximation has an optimal strong convergence rate.

Motivation & Objective

• To establish weak existence of solutions for stochastic McKean-Vlasov equations under non-Lipschitz coefficients.
• To prove pathwise uniqueness of weak solutions under two non-Lipschitz conditions.
• To demonstrate the existence and uniqueness of a strong solution by combining weak existence and pathwise uniqueness.
• To analyze the convergence rate of the Euler-Maruyama approximation for such equations.
• To provide a simpler alternative to complex martingale solution constructions by using direct approximation methods.

Proposed method

• Uses Euler-Maruyama approximation to prove weak existence of solutions under linear growth and non-Lipschitz conditions.
• Applies a fixed-point argument and tightness arguments to establish convergence of approximating sequences.
• Employs the Burkholder-Davis-Gundy (BDG) inequality and Hölder's inequality to control moments of stochastic integrals.
• Introduces a modulus of continuity $ \kappa_\eta $ to handle non-Lipschitz behavior in coefficients.
• Uses Gronwall-type inequalities combined with iterative estimates to bound the difference between exact and approximate solutions.
• Applies Lemma 144 from [10] and results from [13] to derive exponential decay estimates for the approximation error.

Experimental results

Research questions

• RQ1 Can weak solutions exist for stochastic McKean-Vlasov equations when coefficients are non-Lipschitz and satisfy linear growth?
• RQ2 Under what non-Lipschitz conditions is pathwise uniqueness of weak solutions guaranteed?
• RQ3 Does the Euler-Maruyama scheme converge strongly to the true solution under these conditions?
• RQ4 What is the optimal convergence rate of the Euler-Maruyama approximation for such equations?
• RQ5 Can the martingale solution be constructed directly via approximation without complex measure-theoretic machinery?

Key findings

• Weak solutions exist for stochastic McKean-Vlasov equations under linear growth and non-Lipschitz conditions via Euler-Maruyama approximation.
• Pathwise uniqueness holds under two distinct non-Lipschitz conditions, enabling the construction of strong solutions.
• The Euler-Maruyama approximation converges strongly to the true solution with optimal rate $ O(2^{-n}) $.
• The convergence rate is derived using moment estimates, BDG inequality, and iterative Gronwall-type arguments.
• The method avoids intricate calculations used in prior works by directly constructing martingale solutions through approximation.
• The error bound $ \mathbb{E}\left(\sup_{t\in[0,T]}|X_t^n - X_t|^2\right) = O(2^{-n}T_0) $ confirms the optimal convergence rate.