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Convergence rates for Backward SDEs driven by Lévy processes

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.

Convergence rates for Backward SDEs driven by Lévy processes

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 -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.
Paper Structure (10 sections, 10 theorems, 87 equations)

This paper contains 10 sections, 10 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Approximation of Lévy processes
  3. Setting and main results
  4. Setting
  5. Main results
  6. Approximation of the BSDE generator
  7. Optimality
  8. The case $\beta=\beta_*$
  9. Proofs
  10. Approximation of Lévy processes by random walks

Key Result

Lemma 2.1

Let $X$ and $X^n$ be as in levylim and levyapp--can-app-lev-meas and assume that the Blumenthal--Getoor index satisfies $\beta_*<2.$ Then, for any $n\ge 1$ and $\beta\in(\beta_*,2)$, we have the following inequality where $C_\beta = 2 (\int_{\mathbb{R}^d} \left\|x \right\|^\beta \nu(\mathrm d x))^{\frac{1}{2}}$.

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Example 2.3: Generalized Hyperbolic process
  • Example 2.4: CGMY process
  • Example 2.5: Meixner process
  • Example 2.6: Pure-jump Merton model
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • ...and 24 more