The present book deals with a streamlined presentation of Lévy processes and their densities. It is directed at advanced undergraduates who have already completed a basic probability course. Poisson random variables, exponential random variables, and the introduction of Poisson processes are presented first, followed by the introduction of Poisson random measures in a simple case. With these tools the reader proceeds gradually to compound Poisson processes, finite variation Lévy processes and finally one-dimensional stable cases. This step-by-step progression guides the reader into the construction and study of the properties of general Lévy processes with no Brownian component. In particular, in each case the corresponding Poisson random measure, the corresponding stochastic integral, and the corresponding stochastic differential equations (SDEs) are provided. The second part of the book introduces the tools of the integration by parts formula for jump processes in basic settings and first gradually provides the integration by parts formula in finite-dimensional spaces and gives a formula in infinite dimensions. These are then applied to stochastic differential equations in order to determine the existence and some properties of their densities. As examples, instances of the calculations of the Greeks in financial models with jumps are shown. The final chapter is devoted to the Boltzmann equation.
Author(s): Kohatsu-Higa, Takeuchi
Series: Universitext
Publisher: Springer
Year: 2019
Language: English
Pages: 363
Front Matter ....Pages i-xix
Review of Some Basic Concepts of Probability Theory (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 1-7
Front Matter ....Pages 9-9
Simple Poisson Process and Its Corresponding SDEs (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 11-29
Compound Poisson Process and Its Associated Stochastic Calculus (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 31-69
Construction of Lévy Processes and Their Corresponding SDEs: The Finite Variation Case (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 71-100
Construction of Lévy Processes and Their Corresponding SDEs: The Infinite Variation Case (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 101-130
Multi-dimensional Lévy Processes and Their Densities (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 131-143
Flows Associated with Stochastic Differential Equations with Jumps (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 145-154
Front Matter ....Pages 155-155
Overview (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 157-160
Techniques to Study the Density (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 161-172
Basic Ideas for Integration by Parts Formulas (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 173-201
Sensitivity Formulas (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 203-230
Integration by Parts: Norris Method (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 231-267
A Non-linear Example: The Boltzmann Equation (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 269-315
Further Hints for the Exercises (Arturo Kohatsu-Higa, Atsushi Takeuchi)....Pages 317-346
Back Matter ....Pages 347-355
