Stochastic calculus

Stochastic calculus is a special area of mathematics that studies things that change in unpredictable ways. It helps us understand movements, like how tiny particles move in water or how prices in a store go up and down.

This part of math was developed by a Japanese mathematician named Kiyosi Itô during World War II. One famous idea is the Wiener process, named after Norbert Wiener. It helps explain how very small particles move in something called Brownian motion, which scientists Louis Bachelier and Albert Einstein talked about long ago. Today, these ideas are used in money and economics to study how stock prices and interest rates change.

There are different kinds of stochastic calculus, like Itô calculus and Malliavin calculus. These tools help solve problems that are tricky with regular math. They are useful in engineering and other fields where changes happen in complicated ways.

Itô integral

Main article: Itô calculus

The Itô integral is an important idea in stochastic calculus. It helps us add up changes that happen in special kinds of math called stochastic processes. This integral works with certain types of processes and is useful in many areas of science and math.

Stratonovich integral

Main article: Stratonovich integral

The Stratonovich integral is a way to link two special kinds of math ideas. It helps us see how these ideas work together over time. This method was created using another important tool called the Itô integral, 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 thought to change in special ways. These changes follow math rules called stochastic differential equations. One famous example is the Black–Scholes model. This model helps figure out the value of options by treating their prices as if they move like a process called geometric Brownian motion. This shows the chances and risks of 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 about processes that change in unpredictable ways.