Good morning,
I'd like to ask a question about details concerning a part of the proof of the Kolmogorov-Chentsov continuity theorem of stochatic processes (SP), w. l. o. g. we prove it on $[0,1]$. Assumptions of the theorem:
- $(X_t)_{t \geq 0}$ is a SP on $(\Omega, \mathcal{F},\mathbb{P})$.
- There exist $\alpha, \beta, \gamma >0$, s. t. $\mathbb{E}|X_t - X_s|^{\alpha} \leq \gamma |t-s|^{1+\beta}$ for $t,s \in [0,1]$
Definitions:
- $D = \cup _{m \in \mathbb{N}} D_m$ with $D_m = \{ k2^{-m} | k \in \{0, 1, \dots, 2^m\} \}$
The part, which causes problems to my understanding, claims to prove, that $X$ is continous on D.
What we know already (most important steps of the proof):
- We have for $\omega \in \Omega$ except for some set of measure $0$: $|X_{k2^{-n}} (\omega) - X_{(k-1)2^{-n}}(\omega)| \leq 2^{-\mu n}$ for $n \geq n_0 (\omega)$ for $\mu < \frac{\beta}{\alpha}$ we call this $(*)$
- For $m \geq n \geq n_0 (\omega)$ as well as $t, s \in D_m$ with $|t-s| \leq 2^{-n}$ we have: $|X_t (\omega) - X_s (\omega)| \leq \frac{2^{-\mu n}}{1-2^{-\mu}}$ we call this $(**)$
My questions:
- Assume, $t, s \in D$ with $|t-s|\leq 2^{-n_0}$. Then there is some $n \geq n_0$ with $2^{-(n+1)} < |t-s|\leq 2^{-n}$. Then: $|X_t (\omega) - X_s (\omega)| \overset{(**)}{\leq} \frac{2^{-\mu n}}{1-2^{-\mu}} \overset{?}{\leq} \frac{2^\mu |t-s|^\mu}{1-2^{-\mu}}$ (why does this inequality hold?)
- For $t,s \in D$ with $|t-s|>2^{-n_0}$ due to $(*)$ applied on at most $2^{n_0}$ pieces of length $2^{-n_0}$ and $(**)$ applied on one piece of length $<2^{-n_0}$, it is supposed to follow: $|X_t (\omega) - X_s (\omega)| \overset{(*),(**),?}{\leq} 2^{n_0}2^{-\mu n_0} + \frac{2^{-\mu n_0}}{1-2^{\mu}} \overset{?}{\leq} (2^{n_0} + \frac{1}{1-2^{-\mu}})|t-s|^{\mu}$
- We get $|X_t (\omega) - X_s (\omega)| \leq K(\omega) |t-s|^{\mu}$. Why is $\mu$ random?
Best regards