Stochastic Processes

  • Vincenzo Capasso
  • David Bakstein
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


This chapter contains most of the fundamental concepts and results on the general theory of continuous-time stochastic processes. Building on the basic theorem by Kolmogorov–Bochner on the existence of stochastic processes as an extension of finite-dimensional distributions, it is shown that Gaussian processes, processes with independent increments, and Markov processes can be well defined. Continuous-time martingales are introduced in order to present the classical martingale problem as well as in anticipation of the properties of the It integral and solutions of stochastic differential equations. Among key examples, Wiener processes are introduced, followed later on by counting and Poisson processes. The main properties of Wiener processes are proven including their path properties. The Poisson process is defined in a general way so as to allow for a rigorous definition of Lvy processes subsequently. An introduction to random measures, marked point processes, and Lvy processes completes a strong foundation for following chapters.


Marked Poisson Process Independent Increments Wiener Process Continuous-time Martingale Bochner-Kolmogorov Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Vincenzo Capasso
    • 1
  • David Bakstein
    • 2
  1. 1.ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences) and Department of MathematicsUniversity of MilanMilanItaly
  2. 2.ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences)University of MilanMilanItaly

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