Convergence of particles and tree based scheme for singular FBSDEs

Paper: arXiv 2212.11917

Authors: Jean-François Chassagneux, Mohan Yang

Abstract

We study an implementation of the theoretical splitting scheme introduced in [Chassagneux and Yang, 2022] for singular FBSDEs [Carmona and Delarue 2013] and their associated quasi-linear degenerate PDEs. The fully implementable algorithm is based on particles approximation of the transport operator and tree like approximation of the diffusion operator appearing in the theoretical splitting. We prove the convergence with a rate of our numerical method under some reasonable conditions on the coefficients functions. This validates a posteriori some numerical results obtained in [Chassagneux and Yang, 2022]. We conclude the paper with a numerical section presenting various implementations of the algorithm and discussing their efficiency in practice.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on complex theoretical mathematics, establishing convergence proofs for particle and tree-based schemes applied to singular FBSDEs. However, empirical testing is strictly numerical simulation rather than empirical backtesting against financial market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Implement & prove convergence for singular FBSDEs splitting scheme] --> B[Inputs: FBSDE Coefficients, Initial State, PDE Parameters]
    B --> C[Hybrid Numerical Scheme]
    C --> D[Particle Approximation: Transport Operator]
    C --> E[Tree-Like Approximation: Diffusion Operator]
    D & E --> F[Convergence & Rate Proof under Regularity Conditions]
    F --> G[Multi-Asset Numerical Experiments & A Posteriori Validation]