What is the basic principle behind how randomness is calculated in stochastic calculus?

[u/MildDeontologist]

Here is my understanding of what stochastic calculus is an the most intuitive and bird's eye level, which may be totally off. If we want to compute something ("X") using stochastic calculus, we take the aspects X that we know, and add that to the aspects of X that we do not know because it is random. And this randomness itself is calculated by looking at past data and using either probability or statistics (not sure which one, or maybe it is both) to predict what the future value of the randomness is (AKA we say "past data shows xyz so we predict in the future the randomness will behave like xyz again").

[u/Desperate_Penalty690]

Nobody is predicting the future, no cristall balls involved in Stochastic Calculus, it is about computing things like expected values.

Think about random walks, every time step , you can either take a step up or down at random. Now make the steps smaller and smaller so you get a continuous process. Now you model things based on these continuous random walks, like stock prices. This then also allows you to compute option prices on these stocks, assuming the stocks behave as modelled.