Stochastic calculus

Stochastic calculus is a special area of mathematics that works with stochastic processes—systems that change over time in unpredictable ways. It helps mathematicians define and understand integrals, which are tools for adding up tiny pieces, when those integrals involve stochastic processes. This important field was developed 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. This process is used to model Brownian motion, the random movement of tiny particles in liquid or gas, first described by Louis Bachelier in 1900 and later by Albert Einstein in 1905. It also helps scientists study how particles spread out, or diffuse, in space when they are pushed by random forces.

Since the 1970s, stochastic calculus has become very useful in financial mathematics and economics. It helps experts understand how stock prices and interest rates change over time. There are different types of stochastic calculus, such as Itô calculus and Malliavin calculus. Another type, called Stratonovich integral, is often used in engineering. These tools are important because they help solve problems in many different areas of science and math.

Itô integral

Main article: Itô calculus

The Itô integral is a key idea in stochastic calculus, a special kind of math that deals with processes that change in unpredictable ways. It helps us understand how to add up changes that happen randomly over time, which is useful in many areas like physics and finance. This integral works with certain types of processes and helps build a consistent way to handle these tricky calculations.

Stratonovich integral

Main article: Stratonovich integral

The Stratonovich integral is a way to integrate one mathematical process with respect to another in stochastic calculus. It is closely related to the Itô integral, another important tool in this field. This method helps in solving problems involving random changes over time.

Applications

Stochastic calculus is used in mathematical finance to help understand how asset prices change. For example, the Black–Scholes model uses stochastic calculus to price options, treating prices as if they move like a geometric Brownian motion. This shows how stochastic calculus helps in seeing both the chances and risks in financial markets.

Main article: Black–Scholes model

Stochastic integrals

Besides the well-known Itô and Fisk–Stratonovich integrals, there are other ways to understand stochastic integrals. Some of these include the Hitsuda–Skorokhod integral, the Marcus integral, and the Ogawa integral. These different integrals help mathematicians study and solve problems involving random processes.