Eric Forgy
Noncommutative Stochastic Calculus

Given any first order calculus (d,Ω 1(A))(d,\Omega^1(A)) on a commutative algebra AA, we have

d(fg)=d(gf)[df,g]=[dg,f].d(f g) = d(g f)\implies [d f,g] = [d g,f].

According to

this is enough to show that the commutator is a symmetric AA-bilinear function of dfd f and dgd g, i.e.

[df,g]=C(df,dg).[d f,g] = C(d f,d g).

The symmetric AA-bilinear map CC extends to a product

C:Ω 1(A)×Ω 1(A)Ω 1(A),C:\Omega^1(A)\times\Omega^1(A)\to\Omega^1(A),

since Ω 1(A)\Omega^1(A) is generated by elements of the form a(db)a(d b) for a,bAa,b\in A. We will denote this product

αβC(α,β).\alpha\bullet\beta \coloneqq C(\alpha,\beta).

This allows us to write the product rule in “left component form” as

d(fg)=f(dg)+g(df)+dfdg,d(f g) = f(d g) + g(d f) + d f\bullet d g,

in “right component form” as

d(fg)=(dg)f+(df)gdfdg,d(f g) = (d g)f + (d f)g - d f\bullet d g,

or the “symmetrized form” as

d(fg)=12[f(dg)+(dg)f]+12[g(df)+(df)g].d(f g) = \frac{1}{2}\left[f(d g) + (d g)f\right] + \frac{1}{2}\left[g(d f) + (d f)g\right].
Created on January 4, 2010 at 15:23:35 by Eric Forgy