Jump SDEs and the Study of Their Densities

This book consists of two parts: the first part focuses on the stochastic calculus for Levy processes while the second part studies the densities of stochastic differential equations with jumps. 

 

The goal of the second part of the book is to show how to obtain integration by parts (IBP) formulas for random variables which are obtained through systems based on an infinite sequence of independent random variables.  In chapter 9, they describe some techniques to study the density of random variables. Most of these techniques are analytic in nature and they give a different range of results concentrating on the multi-dimensional case. In chapter 10, they present a few basic ideas of how this can be done for stochastic difference equations generated by a finite number of input noises which have smooth densities.

 

Let  \( Z_t \) is a Levy process with Levy measure \( \nu \) that may depend on various parameters. Let  \( G \) be a real-valued bounded measurable function which may also depend on some parameters and is not necessarily smooth.  For many stability reasons, one may be interested in having explicit expressions for the partial derivatives of the \( \mathbb{E}(G(Z_t)) \) with respect to the parameters in the model. These quantities are called Greeks in finance but they may have different names in other fields.  

 

In chapter 11, the authors provide some examples of how to compute these quantities in the case that $G$ is not necessarily differentiable.  In chapter 12, the authors extend the method of analysis introduced in Chap. 11 to a general framework. This method was essentially introduced by Norris to obtain an integration by parts (IBP) formula for jump-driven stochastic differential equations. They focus their study on the directional derivative of the jump measure which respect to the direction of the Girsanov transformation.  The purpose of chapter 13  is to show an application of the concepts of integration by parts introduced before in an explicit example. They have chosen as a model the Boltzmann equation.  Finally, in chapter 14, the authors collect some hints to solve various exercises appearing in this book.

 

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.

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Nikos Halidias is a professor of mathematics at the University of the Aegean, Department of Statistics and Actuarial - Financial Mathematics, Greece. His research and teaching interests are in Differential Equations and Stochastic Analysis.