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A note on stochastic integrals as $L^2$-curves

Stefan Tappe

TL;DR

The paper addresses the problem of linking stochastic integrals viewed as $L^2$-curves with the classical Itô-integral for càdlàg integrands, and extends this connection to Lévy-driven SPDEs. It proves an embedding of the adapted $L^2$-curve space into $L^2(\mathcal{P}_T)$ via a predictable version ${}^p\Phi$ and shows that the Itô-integral ${}^p\Phi \cdot X$ is a càdlàg version of the Onno-Levy integral $(\text{G-})(\Phi \cdot X)$. The results unify integrals with respect to Lévy martingales, Lebesgue measure, and Lévy processes, and relate the integral to predictable projection. An outline SPDE application demonstrates existence and uniqueness of mean-square continuous mild solutions, with càdlàg adapted solutions under a pseudo-contractive semigroup.

Abstract

In a work of van Gaans (2005a) stochastic integrals are regarded as $L^2$-curves. In Filipović and Tappe (2008) we have shown the connection to the usual Itô-integral for càdlàg-integrands. The goal of this note is to complete this result and to provide the full connection to the Itô-integral. We also sketch an application to stochastic partial differential equations.

A note on stochastic integrals as $L^2$-curves

TL;DR

The paper addresses the problem of linking stochastic integrals viewed as -curves with the classical Itô-integral for càdlàg integrands, and extends this connection to Lévy-driven SPDEs. It proves an embedding of the adapted -curve space into via a predictable version and shows that the Itô-integral is a càdlàg version of the Onno-Levy integral . The results unify integrals with respect to Lévy martingales, Lebesgue measure, and Lévy processes, and relate the integral to predictable projection. An outline SPDE application demonstrates existence and uniqueness of mean-square continuous mild solutions, with càdlàg adapted solutions under a pseudo-contractive semigroup.

Abstract

In a work of van Gaans (2005a) stochastic integrals are regarded as -curves. In Filipović and Tappe (2008) we have shown the connection to the usual Itô-integral for càdlàg-integrands. The goal of this note is to complete this result and to provide the full connection to the Itô-integral. We also sketch an application to stochastic partial differential equations.

Paper Structure

This paper contains 6 sections, 8 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Stochastic integrals as $L^2$-curves
  3. Stochastic integral with respect to a Lévy martingale
  4. Stochastic integral with respect to Lebesgue measure
  5. Stochastic integral with respect to a Lévy process
  6. Solutions of stochastic partial differential equations as $L^2$-curves

Key Result

Lemma 2.1

Onno-Levy Let $\Phi \in C_{\rm ad}[0,T]$ be arbitrary. For each $t \in [0,T]$ there exists a unique random variable $Y_t \in L^2(\Omega)$ such that for every $\varepsilon > 0$ there exists $\delta > 0$ such that for every partition $0 = t_0 < t_1 < \ldots < t_n = t$ with $\sup_{i = 0,\ldots,n-1} |t_{i+1} - t_i| < \delta$.

Theorems & Definitions (14)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 4 more