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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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## Notes and References. 1

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## Notes and References.2

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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