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Itô stochastic integral. Itô formula. Tanaka formula

  • Dmytro Gusak
  • Alexander Kukush
  • Alexey Kulik
  • Yuliya Mishura
  • Andrey Pilipenko
Chapter
Part of the Problem Books in Mathematics book series (PBM)

Abstract

Let \(\{W(t),t\in \mathbb{R}^+\}\) be a Wiener process, and \(\{g(t),{\mathcal{F}}_t^W,t\in \mathbb{R}^+\}\) a stochastic process (recall that the previous notation means that g is adapted to a natural filtration \(\{{\mathcal{F}}_t^W\}\) of the Wiener process). Let \({\mathcal{F}}^W=\sigma\{W_t, t\geq 0\}.\) A process g is said to belong to the class \(\hat{\mathcal{L}}_2([a,b])\) if it is measurable and \({{\mathsf{E}}}\int_a^b g^2(s) d s<\infty\). A process g belongs to the class \(\hat{\mathcal{L}}_2\) if it belongs to \(\hat{\mathcal{L}}_2([0,t])\) for all \(t\in \mathbb{R}^+\).

Keywords

Wiener Process Stochastic Integral Local Martingale Integrable Trajectory Natural Filtration 
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.

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Dmytro Gusak
    • 1
  • Alexander Kukush
    • 2
  • Alexey Kulik
    • 1
  • Yuliya Mishura
    • 3
  • Andrey Pilipenko
    • 1
  1. 1.Institute of Mathematics of Ukrainian National Academy of SciencesKyivUkraine
  2. 2.Department of Mathematical Analysis Faculty of Mechanics and MathematicsNational Taras Shevchenko University of KyivKyivUkraine
  3. 3.Department of Probability Theory and Mathematical Statistics Faculty of Mechanics and MathematicsNational Taras Shevchencko University of KyivKyivUkraine

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