Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Poisson-Alekseev-Gröbner formula through Malliavin calculus for Poisson random integrals

Paul Maurer, Jérémy Zurcher

TL;DR

The paper develops a Poisson-driven Alekseev--Gröbner formula that represents the global error between f evaluated at the flow X of an SDE with jumps and at a reference process Y in terms of infinitesimal coefficient discrepancies. The key innovation is a Malliavin-calculus framework for Poisson random measures that yields a Skorohod Poisson integral compatible with extended filtrations, enabling treatment of anticipative terms. The main result (Theorem maintheo) is complemented by an Itô formula for independent random functionals, a detailed construction of smooth random variables, and a rigorous discretisation-limit argument that leverages Vitali-type convergence and weak L^2 convergence. The approach supports weak and strong convergence analysis for jump-diffusion systems with tempered-stable Lévy measures and provides tools for analyzing perturbations and numerical schemes under relatively mild joint-continuity assumptions. Overall, the work advances perturbation analysis and approximation of Poisson-driven SDEs by connecting infinitesimal coefficient errors to global functional errors through a robust Malliavin-calculus framework.

Abstract

In this paper, we establish an Alekseev--Gröbner formula for stochastic differential equations (SDEs) driven by a Poisson random measure, which express the global error between a functional of two processes solution of SDEs started at the same initial condition, in terms of the infinitesimal error (i.e, the difference between the SDEs coefficients). In particular, we consider the situation where the flow process is only assumed to be jointly stochastically continuous with respect to space and time. Our proof relies on a new approach for the definition of the Skorohod--Poisson integral to treat the anticipating term appearing in the formula, based on a definition of the Malliavin derivative on a class of smooth random variables, instead of the more standard polynomial chaos approach.

A Poisson-Alekseev-Gröbner formula through Malliavin calculus for Poisson random integrals

TL;DR

The paper develops a Poisson-driven Alekseev--Gröbner formula that represents the global error between f evaluated at the flow X of an SDE with jumps and at a reference process Y in terms of infinitesimal coefficient discrepancies. The key innovation is a Malliavin-calculus framework for Poisson random measures that yields a Skorohod Poisson integral compatible with extended filtrations, enabling treatment of anticipative terms. The main result (Theorem maintheo) is complemented by an Itô formula for independent random functionals, a detailed construction of smooth random variables, and a rigorous discretisation-limit argument that leverages Vitali-type convergence and weak L^2 convergence. The approach supports weak and strong convergence analysis for jump-diffusion systems with tempered-stable Lévy measures and provides tools for analyzing perturbations and numerical schemes under relatively mild joint-continuity assumptions. Overall, the work advances perturbation analysis and approximation of Poisson-driven SDEs by connecting infinitesimal coefficient errors to global functional errors through a robust Malliavin-calculus framework.

Abstract

In this paper, we establish an Alekseev--Gröbner formula for stochastic differential equations (SDEs) driven by a Poisson random measure, which express the global error between a functional of two processes solution of SDEs started at the same initial condition, in terms of the infinitesimal error (i.e, the difference between the SDEs coefficients). In particular, we consider the situation where the flow process is only assumed to be jointly stochastically continuous with respect to space and time. Our proof relies on a new approach for the definition of the Skorohod--Poisson integral to treat the anticipating term appearing in the formula, based on a definition of the Malliavin derivative on a class of smooth random variables, instead of the more standard polynomial chaos approach.

Paper Structure

This paper contains 37 sections, 253 equations.

Table of Contents

  1. Introduction
  2. Main result and motivations
  3. Method of proof
  4. Treatment of anticipating integrands with extended filtrations.
  5. Malliavin calculus for Poisson random integrals.
  6. Sketch of the proof of Theorem \ref{['maintheo']}
  7. On the space-time joint continuity of the flow.
  8. Organization of the paper
  9. The Poisson--Alekseev--Gröbner formula
  10. Hypotheses
  11. On the coefficients and test functions
  12. On the flows and processes
  13. On the Lévy measure
  14. Main result
  15. A simple example
  16. ...and 22 more sections