Erschienen in:

2001 | OriginalPaper | Buchkapitel

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

verfasst von : Robert S. Liptser, Albert N. Shiryaev

Erschienen in: Statistics of Random Processes

Verlag: Springer Berlin Heidelberg

Enthalten in: Professional Book Archive

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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) 4.1$$M\left( {{\beta _t}|F_s^\beta } \right) = {\beta _s},t \geqslant s $$4.2$$M\left[ {{{\left( {{\beta _t} - {\beta _s}} \right)}^2}|F_s^\beta } \right] = t - s,t \geqslant s. $$

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