Convergence rates for Backward SDEs driven by Lévy processes

Paper: arXiv 2402.01337

Authors: Chenguang Liu, Antonis Papapantoleon, Alexandros Saplaouras

Abstract

We consider Lévy processes that are approximated by compound Poisson processes and, correspondingly, BSDEs driven by Lévy processes that are approximated by BSDEs driven by their compound Poisson approximations. We are interested in the rate of convergence of the approximate BSDEs to the ones driven by the Lévy processes. The rate of convergence of the Lévy processes depends on the Blumenthal–Getoor index of the process. We derive the rate of convergence for the BSDEs in the $\mathbb L^2$-norm and in the Wasserstein distance, and show that, in both cases, this equals the rate of convergence of the corresponding Lévy process, and thus is optimal.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical analysis of BSDEs driven by Lévy processes, establishing optimal convergence rates using advanced stochastic calculus tools like the Blumenthal-Getoor index and Wasserstein distances. However, it is entirely pure mathematics with no empirical validation, real data backtesting, or computational implementations provided.

Research Flowchart

  flowchart TD
    A[Research Goal: Determine optimal BSDE convergence rate<br/>driven by approximated Lévy processes] --> B[Input: Target Lévy Process & BSDE<br/>with Blumenthal-Getoor Index]
    B --> C[Method: Construct Compound Poisson Approximation<br/>for the driving Lévy process]
    C --> D[Method: Formulate Approximate BSDEs<br/>driven by Compound Poisson processes]
    D --> E[Analysis: Derive L2-norm & Wasserstein<br/>distance error bounds]
    E --> F[Outcome: BSDE convergence rate matches<br/>Lévy process rate, proving optimality]