Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on stochastic integration theory with respect to càdlàg semimartingales and a brief introduction to Lévy processes

Mark van den Bosch

TL;DR

The notes provide a practical, proof-light overview of stochastic integration for càdlàg semimartingales, emphasizing the canonical decomposition and the extension of integration theory beyond the continuous setting. They establish core tools—Lebesgue–Stieltjes integration, quadratic variation, compensators, and Doléans measures—and connect them to Lévy processes via Lévy–Khintchine and Lévy–Itô frameworks. The text then explores stochastic differential equations with jumps, including existence/uniqueness results and numerical schemes for simulating Lévy-driven dynamics and stochastic delay equations. Together, these insights enable robust modeling of systems with discontinuities and stochastic jumps, with practical guidance for simulation and analysis.

Abstract

The purpose of these notes is to distribute, mostly without proofs, fundamental definitions and results concerning the theory of semimartingales and stochastic integration. The material serves as a foundational guide for those interested in applying these concepts, particularly in the study of stochastic (functional) differential equations driven by Lévy processes. These notes are adapted from the preliminary chapter of the author's master's thesis (with only minor changes) and are intended to introduce newcomers to the essentials of càdlàg semimartingale theory while also discussing the advantages, limitations, and subtleties as compared to stochastic integration in the continuous setting.

Notes on stochastic integration theory with respect to càdlàg semimartingales and a brief introduction to Lévy processes

TL;DR

The notes provide a practical, proof-light overview of stochastic integration for càdlàg semimartingales, emphasizing the canonical decomposition and the extension of integration theory beyond the continuous setting. They establish core tools—Lebesgue–Stieltjes integration, quadratic variation, compensators, and Doléans measures—and connect them to Lévy processes via Lévy–Khintchine and Lévy–Itô frameworks. The text then explores stochastic differential equations with jumps, including existence/uniqueness results and numerical schemes for simulating Lévy-driven dynamics and stochastic delay equations. Together, these insights enable robust modeling of systems with discontinuities and stochastic jumps, with practical guidance for simulation and analysis.

Abstract

The purpose of these notes is to distribute, mostly without proofs, fundamental definitions and results concerning the theory of semimartingales and stochastic integration. The material serves as a foundational guide for those interested in applying these concepts, particularly in the study of stochastic (functional) differential equations driven by Lévy processes. These notes are adapted from the preliminary chapter of the author's master's thesis (with only minor changes) and are intended to introduce newcomers to the essentials of càdlàg semimartingale theory while also discussing the advantages, limitations, and subtleties as compared to stochastic integration in the continuous setting.
Paper Structure (22 sections, 46 theorems, 205 equations, 7 figures)

This paper contains 22 sections, 46 theorems, 205 equations, 7 figures.

Table of Contents

  1. Stochastic integration
  2. Prerequisites regarding the definition of a semimartingale
  3. Stochastic integrals with respect to semimartingales
  4. Quadratic variation processes and compensators
  5. Stochastic integrals with respect to $L^2$-martingales: feasible extensions on the class of integrands
  6. Other relevant notions and machinery
  7. A brief tutorial on Lévy processes
  8. Examples, illustrations, and the Lévy--Itô decomposition
  9. A special class: regulated Lévy martingale processes
  10. Stochastic differential equations and simulations
  11. MATLAB codes
  12. Main programme (driver.m)
  13. Elementary functions
  14. Brownian motion with drift
  15. Poisson process
  16. ...and 7 more sections

Key Result

Proposition 1.1

Suppose $X=(X_t)_{t\geqslant 0}$ and $Y=(Y_t)_{t\geqslant 0}$ are two stochastic processes, with $X$ a version of $Y.$ If $X$ and $Y$ have right-continuous paths $\mathbb P$-a.s., or if $X$ and $Y$ have left-continuous paths $\mathbb P$-a.s., then $X$ and $Y$ are indistinguishable.

Figures (7)

  • Figure 1: Five sample path realisations of the one-dimensional Lévy process $L=(L_t)_{t\geqslant 0}$, given by $L_t=b t+\sigma W_t$ with $b=1$, $\sigma =2$ and where $W=(W_t)_{t\geqslant 0}$ is a standard Brownian motion.
  • Figure 2: On the left, we see ten sample path realisations of the stochastic integral in equation \ref{['eq:SI']}, simulated by the integral approximation scheme in Appendix \ref{['matlab-WdW']}. On the right, we plot the absolute difference between the latter and the direct numerical approximation of the stochastic process $\frac{1}{2}B_t^2-\frac{1}{2}t$ with $\Delta=10^{-6}.$ (We notice that (much) larger choices for $\Delta$ give reasonable results too.)
  • Figure 3: Five sample path realisations of a Poisson process $N=(N_t)_{t\geqslant 0}$ with intensity $\lambda=1$.
  • Figure 4: Five sample path realisations of the process $C=(C_t)_{t\geqslant 0}$, a compound Poisson process given by $C_t=\sum_{k=1}^{N_t}Z_k$ with $N=(N_t)_{t\geqslant 0}$ a Poisson process with intensity $\lambda=1$ and $Z_1\sim \mathcal{N}(0,\sigma^2)$, $\sigma=2$.
  • Figure 5: Five sample path relations of a linear combination of (in fact, the sum of) the processes $L$ and $C$ from Figures \ref{['fig:drift']} and \ref{['fig:compound']} respectively.
  • ...and 2 more figures

Theorems & Definitions (126)

  • Proposition 1.1: Theorem I.2 of book:protter
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • proof
  • ...and 116 more