Itô’s stochastic calculus: Its surprising power for applications

We trace Itô’s early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Itô’s formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Itô’s formula in mathematical finance in the 1970s. Throughout the paper, we treat Itô’s jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions.