In d-dimensions, Ito's Lemma reads
Note that, unlike in the specific case below, the dynamics of the process X is NOT specified.Specific Case
In one-dimension and assuming
Ito's Lemma reads
Notes: Partial f partial t comes from the fact the f is defined as f(t,X). cf. the general case above, where f=f(X). This so-called one-dimensional case is in fact 2-dimensional, with one of the dimension being deterministic (time).Ito's Product Rule
Suppose X and Y are processes driven by the same Brownian motion. Then
where d[X,Y]=dXdY.Note: This can be easily derived by considering a 2-dimensional Ito's Lemma, where f(X,Y)=XY.
Source of formulae images:
http://en.wikipedia.org/wiki/It%C5%8D_calculus
http://en.wikipedia.org/wiki/It%C5%8D%27s_lemma
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