My teacher said it was because of the symmetry but I don't really understand ): somebody can help me?
2026-03-27 20:25:53.1774643153
If Bt is a Brownian movement. Show that for any t and s, P (Bt> Bs) = 1/2.
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$B_t > B_s$ if and only if $B_t - B_s >0 $. Hence, $\mathbb{P}(B_t - B_s >0) = \mathbb{P}(B_t > B_s)$. Now, $B_t - B_s$ has distribution $\mathcal{N}(0,t-s)$, (by definition of Brownian motion) which is symmetric around the origin.