Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Limit theorems for compensated weighted sums and application to numerical approximations

Yanghui Liu

Abstract

In this paper, we consider a "compensated" random sum that arises from numerical approximation of stochastic integrations and differential equations. We show that the compensated sum exhibits some surprising cancellations among its components, a property which allows to transform it into a Skorohod-type Riemann sum. We then establish limit theorem for the compensated sum based on study of the Skorohod-type Riemann sum. Our proof employs techniques from Malliavin calculus and rough path. We apply our limit theorem result to the Euler approximation method for stochastic integrals and additive stochastic differential equations, filling a notable gap in this area of research. We show that the Euler method converges to the solution at the rate $(1/n)^{H+1/2}$, and that this rate is exact in the sense that the asymptotic error distribution solves a linear differential equation.

Limit theorems for compensated weighted sums and application to numerical approximations

Abstract

In this paper, we consider a "compensated" random sum that arises from numerical approximation of stochastic integrations and differential equations. We show that the compensated sum exhibits some surprising cancellations among its components, a property which allows to transform it into a Skorohod-type Riemann sum. We then establish limit theorem for the compensated sum based on study of the Skorohod-type Riemann sum. Our proof employs techniques from Malliavin calculus and rough path. We apply our limit theorem result to the Euler approximation method for stochastic integrals and additive stochastic differential equations, filling a notable gap in this area of research. We show that the Euler method converges to the solution at the rate , and that this rate is exact in the sense that the asymptotic error distribution solves a linear differential equation.
Paper Structure (12 sections, 17 theorems, 303 equations)

This paper contains 12 sections, 17 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminary results
  4. Elements of rough paths
  5. Elements of Malliavin calculus
  6. Some lemmas
  7. Limit theorem for compensated weighted sums
  8. Limit theorem for Skorokhod-type sums
  9. Compensated weighted sums
  10. Euler method for additive SDEs
  11. Rate of convergence
  12. Asymptotic error distribution

Key Result

Theorem 1.1

Let $\ell\in\mathbb{N}$ be such that $\ell H>1/2$. Then the compensated sum in e.cws converges stably to the conditional Gaussian random variable $c_{H}^{1/2}T^{H+1/2}\int_{0}^{T}y_{u}dW_{u}$ as the mesh size of ${\mathcal{P}}_{n}$ tends to zero, where $W$ is a Brownian motion independent of $x$ and

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 25 more