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The Itô Integral

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An Introduction to Continuous-Time Stochastic Processes

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Abstract

Using the introduction to Wiener processes, the Itô integral is defined, and all its main properties are proven. Then the stochastic differential is introduced, and Itô’s formula is proven. Major results from the Itô calculus, including the fundamental martingale representation theorem, are presented. Finally, an introduction to the Itô-Lévy calculus with respect to Lévy processes is introduced up to a generalization of Itô’s formula.

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Notes

  1. 1.

    For a revision, see the appendix A or, in addition, e.g., Kolmogorov and Fomin (1961).

  2. 2.

    For this classical result of analysis, see, e.g., Kolmogorov and Fomin (1961).

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Authors and Affiliations

  1. ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences) and Department of Mathematics, University of Milan, Milan, Italy

    Vincenzo Capasso

  2. ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences), University of Milan, Milan, Italy

    David Bakstein

Authors
  1. Vincenzo Capasso
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  2. David Bakstein
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© 2012 Springer Science+Business Media New York

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Capasso, V., Bakstein, D. (2012). The Itô Integral. In: An Introduction to Continuous-Time Stochastic Processes. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8346-7_3

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