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 However, this idealization comes at some cost: traditional calculus is no longer valid and you have to use the so-call Itô calculus.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:

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5 Responses

  1. Artur Adib says:

    Hola Esteban,

    Very nice set of notes. Coincidentally, I was preparing a 2-3 page note on Ito v. Stratononich, in honor of Prof. Ito for my Blog.

    Best,
    -Artur

  2. Artur Adib says:

    Wait, how did your Blog find this picture of mine??

  3. admin says:

    I don’t know!. Maybe you have it in your wordpress profile.

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