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)
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Notes and References. 1
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Liptser, R.S., Shiryaev, A.N. (2001). The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations. In: Statistics of Random Processes. Applications of Mathematics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-13043-8_5
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