Ito stochastic calculus

Ito stochastic calculus is a branch of mathematics that extends calculus to stochastic processes, which are phenomena that evolve over time in a random manner. It was developed by Japanese mathematician Kiyosi Itō.

A key concept in this field is Brownian motion, a model for the irregular movement of particles suspended in a fluid. This process, first mathematically described by Albert Einstein, is a type of stochastic process and a Markov process.

Ito calculus provides tools to differentiate and integrate with respect to such random processes, which are not differentiable in the traditional sense. This is crucial for modeling systems in finance, physics, and engineering where randomness plays a significant role.12