# Lévy Processes and Stochastic Calculus

@inproceedings{Applebaum2004LvyPA, title={L{\'e}vy Processes and Stochastic Calculus}, author={David Applebaum}, year={2004} }

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These…

## 2,603 Citations

### Introduction to the Mathematics of Lévy Processes

- Mathematics
- 2005

The goal of this sequel is to provide the foundations of the mathematics of Levy processes for the readers with undergraduate knowledge of stochastic processes as simple as possible. The simplicity…

### Introductory Lectures on Fluctuations of Lévy Processes with Applications

- Mathematics
- 2006

Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance…

### Lévy Processes, Phase-Type Distributions, and Martingales

- Mathematics
- 2014

Lévy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance, etc.; apart from Brownian motion and compound…

### Lévy processes and filtering theory

- Mathematics
- 2014

Stochastic filtering theory is the estimation of a continuous random system given a sequence of partial noisy observations, and is of use in many different financial and scientific areas. The main…

### Introduction to Lévy Processes

- Mathematics
- 2013

As continuous-time analogs of random walks, Levy processes form a rich class of stochastic processes, such as Brownian motions, Poisson processes, and Gamma processes. In this introductory note, we…

### Signal processing with Lévy information

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2013

A theory of Lévy information processes, which have stationary independent increments, is developed, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverseGaussian and normal inverse Gaussian type.

### Fluctuation theory and stochastic games for spectrally negative Lévy processes

- Mathematics
- 2007

Levy processes have stationary, independent increments. This seemingly unassuming (defining) property leads to a surprisingly rich class of processes which appear in a large number of applications…

### A Comparison of Two Settings for Stochastic Integration with Respect to Lévy Processes in Infinite Dimensions

- Mathematics
- 2018

We review two settings for stochastic integration with respect to infinite dimensional Levy processes. We relate notions of stochastic integration with respect to square-integrable Levy martingales,…

### Linear hyperfinite Lévy integrals

- Mathematics
- 2011

This article shows that the nonstandard approach to stochastic integration with respect to (C^2 functions of) Levy processes is consistent with the classical theory of pathwise stochastic integration…

### Numerical Approximation of Stochastic Differential Equations Driven by Levy Motion with Infinitely Many Jumps

- Mathematics
- 2015

In this dissertation, we consider the problem of simulation of stochastic differential equations driven by pure jump Levy processes with infinite jump activity. Examples include, the class of…

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In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Levy white noise. As an example we use this theory to…

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In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to alpha-stable Levy motion. We prove an appropriate…

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CHAPTER 1. BROWNIAN MOTION Definition and Construction Markov Property, Blumenthal's 0-1 Law Stopping Times, Strong Markov Property First Formulas CHAPTER 2. STOCHASTIC INTEGRATION Integrands:…