The Wiener process, the stochastic integral over the Wiener process, and stochastic differential equations

  • R. S. Liptser
  • A. N. Shiryayev
Part of the Applications of Mathematics book series (SMAP, volume 5)


Let (Ω, , 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 t β = σ{ω: β s , st}. Then, according to (1.30) and (1.31), (P-a.s.)
$$ M({\beta _t}|\mathcal{F}_s^\beta ) = {\beta _s},{\text{ t}} \geqslant {\text{s}} $$
$$ M[{({\beta _t} - {\beta _s})^2}|\mathcal{F}_s^\beta ] = t - s,{\text{ t}} \geqslant s $$


Weak Solution Stochastic Differential Equation Strong Solution Simple Function Wiener Process 
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Notes and references

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Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • R. S. Liptser
    • 1
  • A. N. Shiryayev
    • 2
  1. 1.Institute for Problems of Control TheoryMoscowUSSR
  2. 2.Institute of Control SciencesMoscowUSSR

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