We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

생체의 모델링과 Simulation

Introduction To Stochastic Calculus With Applications

Mathematics and Computers in Simulation

From the very first results, the mathematical theory of financial markets has undergone several changes, mostly due to financial crises who forced the mathematical-economical community to change the basic assumptions on which the whole theory is founded. Consequently a new mathematical foundation were needed. In particular, the 2007/2008 credit crunch showed the word that a new financial theoretical framework was necessary, since several empirical evidences emerged that aspects that were neglected prior to these years were in fact fundamental if one has to deal with financial markets. The goal of the present thesis goes in this direction; we aim at developing rigorous mathematical instruments that allow to treat fundamental problems in modern financial mathematics. In order to do so, the talk is thus divided into three main parts, which focus on three different topics of modern financial mathematics. 
The first part is concerned with delay equations. In particular, we will prove Feynman-Kac type result for BSDE's with time-delayed generator, as well as an ad hoc Ito formula for delay equations with jumps. The second part deal with infinite dimensional analysis and network models, focusing in particular on existence and uniqueness results for infinite dimensional SPDE's on networks with general non-local boundary conditions. The last part treats the topic of rigorous asymptotic expansions, providing a small noise asymptotic expansion for SDE with LA©vy noise with several concrete application to financial models.

Francesco Giuseppe Cordoni

Papers

Delayed Forward-Backward stochastic PDE's driven by non Gaussian Lévy noise with application in finance