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Concept
Stochastic differential equation
Summary
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations.
SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale. However, other types of random behaviour are possible, such as jump processes like Lévy processes or semimartingales with jumps. Random differential equations are conjugate to stochastic differential equations. Stochastic differential equations can also be extended to differential manifolds.
Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski, although Louis Bachelier was the first person credited with modeling Brownian motion in 1900, giving a very early example of Stochastic Differential Equation now known as Bachelier model. Some of these early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force.
The mathematical theory of stochastic differential equations was developed in the 1940s through the groundbreaking work of Japanese mathematician Kiyosi Itô, who introduced the concept of stochastic integral and initiated the study of nonlinear stochastic differential equations. Another approach was later proposed by Russian physicist Stratonovich, leading to a calculus similar to ordinary calculus.
The most common form of SDEs in the literature is an ordinary differential equation with the right hand side perturbed by a term dependent on a white noise variable. In most cases, SDEs are understood as continuous time limit of the corresponding stochastic difference equations.
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