Stochastic calculus

Stochastic calculus is a special area of mathematics that works with something called stochastic processes. These processes help us understand things that change in unpredictable ways, like how particles move in water or how prices in a store might go up and down.

This field of math was created by a Japanese mathematician named Kiyosi Itô during World War II. One of the most famous stochastic processes is the Wiener process, named after Norbert Wiener. It helps us model how tiny particles move in a process called Brownian motion, which was described by scientists Louis Bachelier and Albert Einstein many years ago. Today, this idea is also used in money and economics to understand how stock prices and interest rates change over time.

There are different types of stochastic calculus, such as Itô calculus and Malliavin calculus. These tools help solve problems that are hard to understand using normal math. They are especially useful in engineering and other areas where changes happen in complex ways.

Itô integral

Main article: Itô calculus

The Itô integral is a key idea in the study of stochastic calculus. It helps us understand how to add up changes that happen in a special kind of math called stochastic processes. This integral works with certain types of processes and can be very useful in many areas of science and math.

Stratonovich integral

Main article: Stratonovich integral

The Stratonovich integral is a way to connect two special types of mathematical processes. It helps in understanding how these processes relate to each other over time. This method was developed using another important tool called the Itô integral, which is named after the mathematician Kiyosi Itô.

Applications

Stochastic calculus has many uses, especially in mathematical finance. In this area, the prices of things like stocks and options are often thought to change in ways that follow special math rules called stochastic differential equations. One famous example is the Black–Scholes model, which helps figure out the value of options by treating their prices as if they move like a process called geometric Brownian motion. This shows both the chances and risks that come from using stochastic calculus in real-world situations.

Stochastic integrals

Besides the classical Itô and Fisk–Stratonovich integrals, many other ideas of stochastic integrals exist, such as the Hitsuda–Skorokhod integral, the Marcus integral, and the Ogawa integral. These tools help mathematicians study and solve problems involving processes that change in unpredictable ways.