Stochastic differential equations (SDE) occur where a system described by differential equations is influenced by random noise. Stochastic differential equations are used in finance (interest rate, stock prices, …), biology (population, epidemics, …), physics (particles in fluids, thermal noise, …), and control and signal processing (controller, filtering, …).
SDEs are used to model various phenomena such as stock price diffusions or physical systems subject to thermal fluctuations. Typically, SDEs are often driven by Brownian motion or other continuous martingales. However, other types of random behavior are possible, such as jump processes--for instance a Poisson process.
Early work on SDEs was done to describe Brownian motion in Einstein's famous paper, and at the same time by Smoluchowski. However, one of the earlier works related to Brownian motion is credited to Bachelier (1900) in his thesis 'Théorie de la spéculation'. This work was followed upon by Langevin. Later Itô and Stratonovich put SDEs on more solid mathematical footing.
