Stochastic calculus is a branch of calculus much like how fractional calculus and calculus of variations are branches of calculus studied by mathematicians. At the same time, stochastic calculus is also built upon stochastic processes. Many stochastic calculus intuitions were proposed by the natural sciences for example, the botanist Robert Brown - the name sake of Brownian motion, the physicist Paul Langevin - for the Langevin equation, and several physicists who studied things like heat equations etc. Since the middle of the 20th centure, quantitative finance has been using the same diffusion models to model prices for securities and derivatives. Stochastic calculus provides a mathematical language to capture randomness and structure in modelling real world phenomena.
There are many flavors of stochastic calculus, and the most frequently studied is Ito calculus due to its nice properties. To fully grasp stochastic calculus, one should at the minimum have mathematical maturity in real analysis, measure theoretic probability, and stochastic processes. Proofs in Ito calculus are draw on Taylor’s theorem, modes convergence, and martingale theory.
(This is a WIP, and many topics are skipped as the notes are taken in order of what I am working on recently and what comes to mind.)