Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces

Gergely Bodó, Markus Riedle

Abstract

In this work, we present a comprehensive theory of stochastic integration with respect to arbitrary cylindrical Lévy processes in Hilbert spaces. Since cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach relies on an alternative approach to stochastic integration by decoupled tangent sequences. The space of deterministic integrands is identified as a modular space described in terms of the characteristics of the cylindrical Lévy process. The space of random integrands is described as the space of predictable processes whose trajectories are in the space of deterministic integrands almost surely. The derived space of random integrands is verified as the largest space of potential integrands, based on a classical definition of stochastic integrability. We apply the introduced theory of stochastic integration to establish a dominated convergence theorem.

Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces

Abstract

In this work, we present a comprehensive theory of stochastic integration with respect to arbitrary cylindrical Lévy processes in Hilbert spaces. Since cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach relies on an alternative approach to stochastic integration by decoupled tangent sequences. The space of deterministic integrands is identified as a modular space described in terms of the characteristics of the cylindrical Lévy process. The space of random integrands is described as the space of predictable processes whose trajectories are in the space of deterministic integrands almost surely. The derived space of random integrands is verified as the largest space of potential integrands, based on a classical definition of stochastic integrability. We apply the introduced theory of stochastic integration to establish a dominated convergence theorem.
Paper Structure (9 sections, 26 theorems, 183 equations)

This paper contains 9 sections, 26 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cylindrical Lévy processes
  4. Infinitely divisible measures and their characteristics
  5. The modular space
  6. Stochastic integrals with deterministic integrands
  7. Stochastic integrals with predictable integrands
  8. Construction of the decoupled tangent sequence
  9. Characterisation of random integrands

Key Result

Theorem 2.2

Let $L$ be an $H$-valued Lévy process with characteristics $(b^{\theta}, Q, \lambda)$, and let $(\pi_n)_{n \in \mathbb{N}}$ be a nested normal sequence of partitions of $[s,t]$, where for each fixed $n \in \mathbb{N}$ we have $\pi_n=\left\{s=p_{0,n}<p_{1,n}<...<p_{{N(n)},n}=t\right\}$. If we put $d_

Theorems & Definitions (74)

  • Remark 2.1
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Definition 3.5
  • Remark 3.6
  • ...and 64 more