[Paper Review] Path regularity of coupled McKean-Vlasov FBSDEs

Christoph Reisinger, Wolfgang Stockinger+1 more|arXiv (Cornell University)|Nov 12, 2020

Stochastic processes and financial applications17 references4 citations

TL;DR

This paper establishes 1/2-Hölder continuity in time for solutions to coupled McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) under Lipschitz conditions, using a Lipschitz decoupling field assumption. The result ensures path regularity essential for numerical approximation and convergence analysis of particle systems and time-stepping schemes.

ABSTRACT

This paper establishes Hölder time regularity of solutions to coupled McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). This is not only of fundamental mathematical interest, but also essential for their numerical approximations. We show that a solution triple to a MV-FBSDE with Lipschitz coefficients is 1/2-Hölder continuous in time in the $L^p$-norm provided that it admits a Lipschitz decoupling field. Special examples include decoupled MV-FBSDEs, coupled MV-FBSDEs with a small time horizon and coupled stochastic Pontryagin systems arsing from mean field control problems.

Motivation & Objective

• To establish path regularity of solutions to fully-coupled McKean-Vlasov FBSDEs with general Lipschitz coefficients.
• To address the lack of regularity results for MV-FBSDEs with degenerate diffusion and strong coupling, unlike classical FBSDEs.
• To provide a foundation for numerical analysis, including particle approximations and time-discretization schemes.
• To verify the regularity under practical conditions such as small time horizons and generalized monotonicity.

Proposed method

• Uses a continuation method via a parameterized family of equations (A.1) with parameter λ ∈ [0,1] to prove existence and stability.
• Applies Banach’s fixed point theorem to show contraction in a suitable Banach space of adapted processes.
• Employs a decoupling field assumption to control the time regularity of the solution triple (X, Y, Z).
• Derives moment bounds and path regularity estimates via a priori estimates and Gronwall-type arguments.
• Relies on Malliavin calculus and first variation processes, adapted to the mean field setting despite the lack of standard derivative representations.
• Establishes the key regularity estimates (1.2) and (1.3) under the decoupling field condition.

Experimental results

Research questions

• RQ1 Under what conditions does a solution to a coupled MV-FBSDE exhibit 1/2-Hölder continuous paths in time?
• RQ2 How can path regularity be established for MV-FBSDEs when standard Malliavin derivative representations fail due to mean field dependence?
• RQ3 What conditions ensure the existence of a Lipschitz decoupling field for MV-FBSDEs?
• RQ4 Can path regularity be extended to coupled systems with degenerate diffusion and general Lipschitz coefficients?
• RQ5 How does the time horizon size affect the regularity and solvability of MV-FBSDEs?

Key findings

• Solutions to MV-FBSDEs with Lipschitz coefficients are 1/2-Hölder continuous in time in the L^p-norm when a Lipschitz decoupling field exists.
• The regularity estimates (1.2) and (1.3) hold with constants depending only on p and the initial condition ξ₀.
• The decoupling field condition is satisfied for decoupled MV-FBSDEs, small time horizons, and systems with generalized monotonicity.
• The solution triple (X, Y, Z) satisfies moment bounds uniformly in time, ensuring stability and integrability.
• The method of continuation with a parameter λ ∈ [0,1] proves unique solvability and stochastic stability under the decoupling assumption.
• The results support convergence analysis for particle approximations and time-stepping schemes in mean field control and game problems.