Jump SDEs and the Study of Their Densities

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.

• Jump processes
• Stochastic Calculus
• Calculus of Variations
• Integration by parts
• Densities of random variables
• partial differential equations

“The book is very friendly to the reader because the authors give all necessary instructions for the non-experienced reader that are also useful for any reader … . the book is devoted to very topical issues appearing in such fields as economy, finance, techniques and many others. Therefore, reading this book is both work and pleasure, and the book can be recommended to both undergraduate and graduate students, specialists in probability theory and its applications.” (Yuliya S. Mishura, zbMATH 1447.60001, 2020)

“The book is very well written and presented, and nicely oriented to self-study with many exercises. I strongly recommend this excellent book to all graduate students and researchers interested in stochastic calculus of jump processes or in modeling using jump processes, in any subject.” (Josep Vives, Mathematical Reviews, October, 2020)

“This book is written mainly for advanced undergraduate and graduate students and researchers that are interested in this field and it can bring the reader very soon to a research level. The list of references are complete and guide the researcher to more specific and advanced topics.” (Nikos Halidias, MAA Reviews, August 16, 2020)

• Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Japan

  Arturo Kohatsu-Higa

• Department of Mathematics, Tokyo Woman’s Christian University, Tokyo, Japan

  Atsushi Takeuchi

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