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Fixing a Minor Mistake in the Theory of Stochastic Integration and Differential Equations

Sebastian Rickelhoff, Alexander Schnurr

Abstract

When considering stochastic integration and the theory of stochastic differential equations, P. Protter's textbook \cite{protter} undoubtedly is a main piece of standard literature. Not only is it well-written, but it also contains various profound results regarding these fields. Unfortunately, Theorem 12 of Chapter V, which presents an equivalence to uniform convergence on compacts in probability, is found to be incorrect. Given that numerous important results rely on this theorem, this paper aims to present a corrected version of it.

Fixing a Minor Mistake in the Theory of Stochastic Integration and Differential Equations

Abstract

When considering stochastic integration and the theory of stochastic differential equations, P. Protter's textbook \cite{protter} undoubtedly is a main piece of standard literature. Not only is it well-written, but it also contains various profound results regarding these fields. Unfortunately, Theorem 12 of Chapter V, which presents an equivalence to uniform convergence on compacts in probability, is found to be incorrect. Given that numerous important results rely on this theorem, this paper aims to present a corrected version of it.
Paper Structure (2 sections, 1 theorem, 13 equations)

This paper contains 2 sections, 1 theorem, 13 equations.

Table of Contents

  1. Introduction
  2. The Original and Corrected Statement

Key Result

Theorem 2.2

Let $H^n, H \in \mathbb{D}$. The following statements are equivalent:

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3