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.
