Question

Show that, with probability 1 , the Brownian sample function $X:[0, \infty) \rightarrow \mathbb{R}$ is not monotonic on any interval $[t, u]$.

   Show that, with probability 1 , the Brownian sample function $X:[0, \infty) \rightarrow \mathbb{R}$ is not monotonic on any interval $[t, u]$.

Fractal Geometry: Mathematical Foundations and Applications

Fractal Geometry: Mathematical Foundations and Applications

Kenneth Falconer 2nd Edition

Chapter 16, Problem 5

Instant Answer

Step 1

- $X(t)$ has independent increments. - $X(t)$ has stationary increments. - For any $t > s \geq 0$, $X(t) - X(s) \sim N(0, t-s)$, where $N(0, t-s)$ denotes a normal distribution with mean 0 and variance $t-s$. - $X(t)$ has continuous paths almost surely. Show more…

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Show that, with probability 1 , the Brownian sample function $X:[0, \infty) \rightarrow \mathbb{R}$ is not monotonic on any interval $[t, u]$.

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