Monday, November 11, 2013

differential ^ integral in Ito's formula

See posts on Ito being the most precise possible prediction.

Given dynamics of S is    dS = mu dt + sigma dW  , and given a (process following) a function f() of S,  then, Ito's rule says

    df = df/dS * dS + 1/2 d(df/dS)/dS * (dS)^2

There are really 2 different meanings to d____

- The df/dS term is ordinary differentiation wrt to S, treating S as just an ordinary variable in ordinary calculus.
- The dt term, if present, isn't a differential. All the d__ appearing outside a division (like d_/d__) actually indicates an implicit integral.
** Specifically, The dS term (another integral term) contains a dW component. So this is even more "unusual" and "different" from the ordinary calculus view point.


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