I will use as example a continuous Brownian process, which each possible trial have undefined derivative in time and unbounded variation, but somehow it also looks like there exists a restriction for the next point to be near the previous one in amplitude.
It is possible to the trial function to suddenly jumps up to infinity? If not, Which "thing" is stopping it?
I know that "concentration of meassure" restricts the tails of probabilities distributions behind these processes... I understand that "high" jumps have really low probabilities, but tails are never "completely cut" and a jump up to "infinity" arent avoided by them.
But a few days ago reading about functions inequalities, I found the kalman-rotta inequality which bounds the rate of change of a function: $$\left|f^{\prime}\right|^2 \leq 4\left|f\right|\left|f^{\prime\prime}\right|$$ and I dont know if something similar applies to stochastic processes, making "impossible" to see a "high amplitude" jump between near/contiguous points of a stochastic process trial function.
Hope you can elaborate in your answers... beforehand, thanks you very much.
If the stochastic process is defined to be a continuous process, as the Wiener process does, then it is forbidden to jump to infinity on every possible closed interval because it will became it discontinuous.
In more technical words, every continuous function defined in a close interval will be bounded by the Extreme value theorem, which close the question.
But I would like to add, also that the distribution of a Wiener process is a Normal Distribution that indeed can have an infinity value, but because of the Extreme Value Theorem, this distribution should be take as the distribution of every possible path on the unbounded time interval, been different from the distribution during finite intervals.
For a good approximation of their possible maximum achievable values (not the running maxima), I made a question here proposing a slight modification to the modulus of continuity of a Wiener process, which must be demonstrated mathematically first, but numerically it works good so I invite you to see it and comment about it.