Stochastic Calculus
Introduction
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.
The best-known stochastic process to which stochastic calculus is applied is the Wiener process, which is used for modeling Brownian motion and other random movements, such as stock market fluctuations.
History
The development of stochastic calculus was an unexpected achievement of the 20th century, primarily by the Japanese mathematician Kiyoshi Itō, who was inspired by the earlier work of Andrei Kolmogorov, Joseph L. Doob, and Paul Lévy in the theory of stochastic processes.
Basic Concepts
Stochastic calculus is based on two key concepts: stochastic processes and martingales.
A stochastic process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.
A martingale is a stochastic process that models a fair game, where knowledge of past events never helps predict the mean of the future winnings, and so, on average, we neither win nor lose money. Martingales are a key element in the stochastic calculus, a branch of mathematics that aims to describe random processes.
Stochastic Integrals
One of the key techniques in stochastic calculus is the use of stochastic integrals. The two main types of stochastic integrals are the Itō integral and the Stratonovich integral.
The Itō integral is named after Kiyoshi Itō and is integral with respect to the Wiener process. It is defined differently from the Riemann-Stieltjes integral, as the terms of the integral need to be adapted to the filtration of the Wiener process.
The Stratonovich integral is named after Ruslan L. Stratonovich. It is used in physics and engineering where a heuristic derivation can be given. The Stratonovich integral can be interpreted as the ordinary calculus where the chain rule of differentiation can be used.
Stochastic Differential Equations
Stochastic calculus is also used to derive stochastic differential equations (SDEs). These are differential equations in which one or more of the terms is a stochastic process, thus resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as unstable stock prices or physical systems subject to random shocks.
Applications
Stochastic calculus has been applied to a wide variety of areas, including physics, engineering, and finance. In finance, it is used to model the behavior of asset prices, particularly in the Black-Scholes model for pricing derivatives. In physics, it is used to solve stochastic differential equations, which can model noise in electronic and communication systems, neural networks, etc.