Advertisement

Stochastic Differential Calculus

  • Roberto Mauri
Part of the Soft and Biological Matter book series (SOBIMA)

Abstract

In this chapter, the basic concepts of stochastic integration are explained in a way that is readily understandable also to a non-mathematician (a more formal, yet still understandable, treatment can be found in Gardiner, Handbook of Stochastic Methods, 1985, Sect. 4.3). The fact that Brownian motion, i.e. the Wiener process, is non-differentiable, and therefore requires its own rules of calculus, is explained in Sect. 5.1. In fact, there are two dominating versions of stochastic calculus, each having advantages and disadvantages, namely the Ito stochastic calculus (Sect. 5.2), based on a pre-point discretization rule, named after Kiyoshi Ito, and the Stratonovich stochastic calculus (Sect. 5.3), based on a mid-point discretization rule, developed simultaneously by Ruslan Stratonovich and Donald Fisk. Finally, in Sect. 5.4, we illustrate the main features of Stochastic Mechanics, showing that, by applying the rules of stochastic integration, the evolution of a random variable can be described through the Schrödinger equation of quantum mechanics.

Keywords

Wiener Process Stochastic Calculus Stochastic Mechanic Brownian Force Forward Kolmogorov Equation 
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.

References

  1. 1.
    Adler, L.: Quantum Theory as an Emergent Phenomenon. Cambridge University Press, Cambridge (2004) CrossRefGoogle Scholar
  2. 2.
    Bohm, D.: Phys. Rev. 85, 166 (1952) MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Correggi, M., Morchio, G.: Ann. Phys. 296, 371 (2002) MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Gardiner, G.W.: Handbook of Stochastic Methods. Springer, Berlin (1985) Google Scholar
  5. 5.
    Guerra, F.: Phys. Rep. 77, 263 (1981) MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Guerra, F., Morato, L.M.: Phys. Rev. D 27, 1774 (1983) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Nelson, E.: Phys. Rev. 150, 1079 (1969) ADSCrossRefGoogle Scholar
  8. 8.
    Nelson, E.: Quantum Fluctuations. Princeton University Press, Princeton (1985) zbMATHGoogle Scholar
  9. 9.
    Pavon, M.: J. Math. Phys. 40, 5565 (1999) MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Tessarotto, M., Ellero, M., Nicolini, P.: Phys. Rev. A 75, 012105 (2007) ADSCrossRefGoogle Scholar
  11. 11.
    van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, p. 245. North-Holland, Amsterdam (1981) zbMATHGoogle Scholar
  12. 12.
    Wallstrom, T.C.: Phys. Rev. A 49, 1613 (1994) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Roberto Mauri
    • 1
  1. 1.Department of Chemical Engineering, Industrial Chemistry and Material ScienceUniversity of PisaPisaItaly

Personalised recommendations