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Stochastic Processes

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

Abstract

We commence along the lines of the founding work of Kolmogorov by regarding stochastic processes as a family of random variables defined on a probability space and thereby define a probability law on the set of trajectories of the process. More specifically, stochastic processes generalize the notion of (finite-dimensional) vectors of random variables to the case of any family of random variables indexed in a general set T. Typically, the latter represents “time” and is an interval of \(\mathbb{R}\) (in the continuous case) or \(\mathbb{N}\) (in the discrete case). For a nice and elementary introduction to this topic, the reader may refer to Parzen (1962).

References

  1. Anderson, W.J.: Continuous-Time Markov Chains: An Application-Oriented Approach. Springer, New York (1991)CrossRefGoogle Scholar
  2. Applebaum, D.: Levy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  3. Ash, R.B., Gardner M.F.: Topics in Stochastic Processes. Academic, London (1975)zbMATHGoogle Scholar
  4. Baldi, P.: Equazioni differenziali stocastiche. UMI, Bologna (1984)Google Scholar
  5. Bauer, H.: Probability Theory and Elements of Measure Theory. Academic, London (1981)zbMATHGoogle Scholar
  6. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  7. Billingsley, P.: Probability and Measure. Wiley, New York (1986)zbMATHGoogle Scholar
  8. Bosq, D.: Linear Processes in Function Spaces. Theory and Applications. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  9. Breiman, L.: Probability. Addison-Wesley, Reading, MA (1968)zbMATHGoogle Scholar
  10. Bremaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Heidelberg (1981)CrossRefzbMATHGoogle Scholar
  11. Chung, K.L.: A Course in Probability Theory, 2nd edn. Academic, New York (1974)zbMATHGoogle Scholar
  12. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Wiley-Interscience, New York (1966)Google Scholar
  13. Çynlar, E.: Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs, NJ (1975)Google Scholar
  14. Daley, D., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer, Berlin (1988)zbMATHGoogle Scholar
  15. Daley, D., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Volume II: General Theory and Structure. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  16. Devijver, P.A., Kittler, J.: Pattern Recognition. A Statistical Approach. Prentice-Hall, Englewood Cliffs, NJ (1982)zbMATHGoogle Scholar
  17. Di Nunno, G., Øksendal, B., Proske, F.: Malliavin Calculus for Levy Processes with Applications to Finance. Springer, Berlin/Heidelberg (2009)CrossRefGoogle Scholar
  18. Doob, J.L.: Stochastic Processes. Wiley, New York (1953)zbMATHGoogle Scholar
  19. Dynkin, E.B.: Markov Processes, vols. 1–2. Springer, Berlin (1965)CrossRefzbMATHGoogle Scholar
  20. Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)CrossRefzbMATHGoogle Scholar
  21. Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1971)zbMATHGoogle Scholar
  22. Ghanen, R.G., Spanos, P.D.: Stochastic Finite Elements. A Spectral Approach. Revised Edition. Dover Publication, Mineola, NY (2003)Google Scholar
  23. Gihman, I.I., Skorohod, A.V.: The Theory of Random Processes. Springer, Berlin (1974)Google Scholar
  24. Grigoriu, M.: Stochastic Calculus: Applications to Science and Engineering. Birkhäuser, Boston (2002)CrossRefGoogle Scholar
  25. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Kodansha (1989)zbMATHGoogle Scholar
  26. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer Lecture Notes in Mathematics. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  27. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  28. Kallenberg, O.: Foundations of Modern Probability. Springer, Berlin (1997)zbMATHGoogle Scholar
  29. Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes. Academic, New York (1975)zbMATHGoogle Scholar
  30. Karlin, S., Taylor H.M.: A Second Course in Stochastic Processes. Academic, New York (1981)zbMATHGoogle Scholar
  31. Karr, A.F.: Point Processes and Their Statistical Inference. Marcel Dekker, New York (1986)zbMATHGoogle Scholar
  32. Klenke, A.: Probability Theory. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  33. Klenke, A.: Probability Theory. A Comprehensive Course, 2nd edn. Springer, Heidelberg (2014)Google Scholar
  34. Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg (2014)CrossRefzbMATHGoogle Scholar
  35. Lamperti, J.: Stochastic Processes: A Survey of the Mathematical Theory. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  36. Last, G., Brandt, A.: Marked Point Processes on the Real Line: The Dynamic Approach. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  37. Mercer, J.: Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. Roy. Soc. Lond. 209, 415–446 (1909)CrossRefzbMATHGoogle Scholar
  38. Métivier, M.: Notions fondamentales de la théorie des probabilités. Dunod, Paris (1968)zbMATHGoogle Scholar
  39. Meyer, P.A.: Probabilités et Potentiel. Ilermann, Paris (1966)zbMATHGoogle Scholar
  40. Mikosch, T.: Non-Life Insurance Mathematics, 2nd edn. Springer, Berlin/Heidelberg (2009)CrossRefzbMATHGoogle Scholar
  41. Parzen, E.: Statistical inference on time series by Hilbert space methods,Technical Report no. 23, Statistics Department, Stanford University, Stanford, CA, Jan 1959Google Scholar
  42. Parzen, E.: Stochastic Processes. Holden-Day, San Francisco (1962)zbMATHGoogle Scholar
  43. Protter, P.: Stochastic Integration and Differential Equations. Springer, Berlin (1990). Second Edition 2004Google Scholar
  44. Ramsay, J.O., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer, Berlin (1990). Second Edition 2004Google Scholar
  45. Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, 2nd edn. Springer, New York (2005)Google Scholar
  46. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Heidelberg (1991)CrossRefzbMATHGoogle Scholar
  47. Robert, P.: Stochastic Networks and Queues. Springer, Heidelberg (2003)CrossRefzbMATHGoogle Scholar
  48. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, vol. 1. Wiley, New York (1994)zbMATHGoogle Scholar
  49. Sato, K.I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  50. Schuss, Z.: Theory and Applications of Stochastic Processes: An Analytical Approach. Springer, New York (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Vincenzo Capasso
    • 1
  • David Bakstein
    • 1
  1. 1.ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences)Università degli Studi di MilanoMilanItaly

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