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 However, this idealization comes at some cost: traditional calculus is no longer valid and you have to use the so-call Itô calculus.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

- I prepared a long time ago some notes about SDE in Spanish
- A nicer introduction to SDEs by Prof. Evans

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

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

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