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The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations

  • Robert S. Liptser
  • Albert N. Shiryaev
Chapter
  • 2.1k Downloads
Part of the Applications of Mathematics book series (SMAP, volume 5)

Abstract

Let (Ω, F, P) be a probability space and β = (β t), t ≥ 0, be a Brownian motion process (in the sense of the definition given in Section 1.4). Denote \(F_t^\beta= \sigma \left\{ {\omega :{\beta _s}} \right.,s \leqslant \left. t \right\}\) Then, according to (1.30) and (1.31),(P-a.s)
$$M\left( {{\beta _t}|F_s^\beta } \right) = {\beta _s},t \geqslant s $$
(4.1)
$$M\left[ {{{\left( {{\beta _t} - {\beta _s}} \right)}^2}|F_s^\beta } \right] = t - s,t \geqslant s. $$
(4.2)

Keywords

Weak Solution Stochastic Differential Equation Strong Solution Simple Function Stochastic Integral 
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Bibliography

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Robert S. Liptser
    • 1
  • Albert N. Shiryaev
    • 2
  1. 1.Department of Electrical Engineering SystemsTel Aviv UniversityTel AvivIsrael
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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