Descript 
1 online resource (xix, 355 pages) : illustrations, digital ; 24 cm 

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computer c rdamedia 

online resource cr rdacarrier 

text file PDF rda 
Series 
Universitext, 01725939


Universitext

Note 
Review of some basic concepts of probability theory  Simple Poisson process and its corresponding SDEs  Compound Poisson process and its associated stochastic calculus  Construction of Levy processes and their corresponding SDEs: The finite variation case  Construction of Levy processes and their corresponding SDEs: The infinite variation case  Multidimensional Levy processes and their densities  Flows associated with stochastic differential equations with jumps  Overview  Techniques to study the density  Basic ideas for integration by parts formulas  Sensitivity formulas  Integration by parts: Norris method  A nonlinear example: The Boltzmann equation  Further hints for the exercises 

The present book deals with a streamlined presentation of Levy 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 Levy processes and finally onedimensional stable cases. This stepbystep progression guides the reader into the construction and study of the properties of general Levy 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 finitedimensional 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 
Host Item 
Springer Nature eBook

Subject 
Stochastic differential equations


Levy processes


Probability Theory and Stochastic Processes


Functional Analysis


Partial Differential Equations

Alt Author 
Takeuchi, Atsushi, author


SpringerLink (Online service)

