Stochastic calculus

This field was created and started by the Japanese mathematician Kiyosi Itô during World War II.

Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.

The main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Itô's lemma.

This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than Rn.

For example, the Black–Scholes model prices options as if they follow a geometric Brownian motion, illustrating the opportunities and risks from applying stochastic calculus.