Studying two concrete examples of stochastic process, which are the Poisson and Wiener process, I noticed that the costruction of process relies on the 'hope' to create a process taking a sort of modified derivative, i.e the nearest things which resembles the idea of increments $\frac{X_t-X_s}{t-s}$ but it doesn't really say why, there is a relatevely simple heuristic or deeper meaning of this fact?
I thought it could be related to the fact that for example, in Wiener process, our request is to create a process which has "limited Hölder increments" in mean through the following :
Theorem : Given $(X_t)_{t \in [0,1]}$ a stochastic process such that exists $\alpha,\beta,c > 0: \mathbb{E}[\lvert X_t -X_s \rvert^{\alpha}] \leq \lvert t-s \rvert^{1+\beta} \hspace{0.1cm} \forall \hspace{0.1cm}t,s \in [0,1] \Rightarrow \hspace{0.1cm} \exists (Y_t)_{t \in [0,1]}$ modification of the process with $\gamma$- Hölder continuos realizations $\forall \hspace{0.1cm} \gamma < \frac{\beta}{\alpha}$
There is an intuitive interpretation of the Kolmogorov continuity lemma applied to Brownian motion on $[0,1]$? I understand why we use the theorem (basically asking for regularity of the realizations) but I don't know wether there's a simple way of understanding heuristically the fact.
Ps : But why bother ? What's the real strength on having the continuity of the realizations ? Because from the picture I linked it's not very clear to me.
