Problem regarding the proof of sample-continuity

Reference no: EM13990694

Let xt be a Brownian motion. Show that for any ε> 0 and almost all ω there is an M (ω) ∞ such that for all s and t in [0, 1], |xs - xt |≤ M (ω)|s - t |0.5-ε . Hint: First, in the proof of sample-continuity (12.1.5), replace 1/n2 by 1/2n(1-ε)/2. Note that these numbers form a geomet- ric series in n, and that if the conclusion holds for each ω for |s - t | small enough, then it holds for all s and t in [0, 1] with perhaps a larger M (ω).

Reference no: EM13990694

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