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Introduction to stochastic differential equations

Stochastic differential equations (SDEs) are basically inhomogenous ordinary differential equations that depend on an external stochastic process.

Typically, that stochastic process is white noise, which is the mathematical idealization of the noise found in nature. This idealization is handy, because it simplifies the mathematical description. However, this idealization comes at some cost: traditional calculus is no longer valid and you have to use the so-call Itô calculus. This introduces some non intuitive changes. For example, instead of the usual chain rule of calculus, the Itô formula should be used. Here is an example, the Itô integral of the Wiener process (or Brownian Motion) is

Note the last term in the right-hand-side (!).

If you are interested to learn more on SDEs:

Author

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Professor at Universidad Carlos III de Madrid and MIT Medialab. Working on Complex Systems, Social Networks and Urban Science.