[Paper Review] Numerical Method for FBSDEs of McKean-Vlasov Type

Jean-François Chassagneux, Dan Crisan+1 more|arXiv (Cornell University)|Mar 6, 2017

Stochastic processes and financial applications3 references3 citations

TL;DR

This paper presents a novel numerical scheme for solving forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type, which arise in mean field control and mean field game problems. The method uses a recursive local Picard iteration on small time intervals, establishing convergence under mild regularity conditions on the decoupling field, with error bounds derived via stochastic Taylor expansions and martingale estimates.

ABSTRACT

This paper is dedicated to the presentation and the analysis of a numerical scheme for forward-backward SDEs of the McKean-Vlasov type, or equivalently for solutions to PDEs on the Wasserstein space. Because of the mean field structure of the equation, earlier methods for classical forward-backward systems fail. The scheme is based on a variation of the method of continuation. The principle is to implement recursively local Picard iterations on small time intervals. We establish a bound for the rate of convergence under the assumption that the decoupling field of the forward-bakward SDE (or equivalently the solution of the PDE) satisfies mild regularity conditions. We also provide numerical illustrations.

Motivation & Objective

• To develop a robust numerical scheme for forward-backward SDEs with mean field (McKean-Vlasov) structure, where classical methods fail due to non-Markovian and non-local dependence on the law of the solution.
• To address the challenge of solving non-local PDEs on Wasserstein space arising in mean field games and control, by linking them to FBSDEs via the decoupling field.
• To establish convergence rates for the proposed scheme under mild regularity assumptions on the decoupling field and coefficients.
• To provide a probabilistic numerical approach that avoids the complexity of traditional PDE solvers on infinite-dimensional measure spaces.

Proposed method

• The scheme employs a variation of the method of continuation, recursively applying local Picard iterations over small time intervals to stabilize the solution of the coupled FBSDE system.
• The forward component is discretized using an Euler-type scheme with time partitioning, while the backward component is approximated iteratively using conditional expectations.
• Stochastic Taylor expansions are used to analyze the error, with key terms involving second-order derivatives of the decoupling field and Wasserstein derivatives in the measure argument.
• Martingale increments are introduced to control the error in the backward component, with bounds derived using the Bürkholder-Davis-Gundy inequality and moment estimates.
• The method leverages the connection between the McKean-Vlasov FBSDE and the master equation (non-local PDE on Wasserstein space), using the decoupling field as the solution proxy.
• A continuous-time version of the scheme is constructed and analyzed, with error controlled via discrete Gronwall-type inequalities and moment bounds on the solution components.

Experimental results

Research questions

• RQ1 Can a numerical scheme be designed for McKean-Vlasov FBSDEs that overcomes the failure of classical methods due to mean field dependence and non-locality in the measure argument?
• RQ2 What is the convergence rate of such a scheme under mild regularity assumptions on the decoupling field and coefficients?
• RQ3 How can the non-Markovian and measure-dependent structure of the equation be handled numerically without resorting to full PDE discretization on the Wasserstein space?
• RQ4 Can recursive local Picard iterations on small time intervals yield stable and convergent approximations for the entire time horizon?
• RQ5 What are the precise moment and error bounds for the scheme, particularly in terms of time step size and regularity of the solution?

Key findings

• The proposed scheme achieves convergence under mild regularity conditions on the decoupling field, with a rate bounded by a constant times the square root of the maximum time step size.
• The error in the backward component is controlled via a martingale increment decomposition, with the expected squared norm of the error term bounded by $ Ch_i^{1/2} $, ensuring stability.
• The forward component's solution norm is bounded by a constant depending on the initial data, the time step, and the error in the decoupling field approximation.
• The scheme's convergence is established through a discrete Gronwall inequality applied to the moment estimates of the solution components.
• Numerical illustrations confirm the theoretical convergence rates and demonstrate the scheme's robustness in practice.
• The method successfully handles the non-local and measure-derivative terms in the master equation by embedding them into the iterative Picard scheme via law-dependent coefficients.