ABSTRACT
We continue to consider different types of random processes, keeping the notation from Chapter 5.
In this section, we revisit Brownian motion, or the Wiener process, wt defined in Section 5.1.2.2.
1.1.1 Non-differentiability of trajectories
Consider a time interval Δ = (t, t + δ]. To determine whether wt has a derivative at t, we should consider the limiting behavior of the r.v. ηΔ = wΔ/δ as δ → 0. By definition, the r.v. wΔ is normal with zero mean and standard deviation
√ δ. Then for x ≥ 0, we have
P(|ηΔ|> x) = P(|wΔ|> xδ) = 2(1−Φ(xδ/ √
δ)) = 2(1−Φ(x √
δ))→ 2(1−Φ(0)) = 1 as δ→ 0. Since the last relation is true for an arbitrary large x, this means that for small values of δ, the r.v. ηΔ takes on arbitrary large values with a probability close to one. Rigorously, ηΔ → ∞ as δ → 0 in probability (for a rigorous definition of this type of convergence see Section 0.5).
