## One forms: link between stochastic integral d and Riemann-Stieltjes d and one form d.

In differential geometry we learn about the "one forms," $dx_{i}$ and its dual $\frac{\partial}{\partial x_{i}}$

I don't see in the literature anything concerning a "one form" of Stieltjes type $df\left(x\right)$.

Is it appropriate to call it a one form? Furthermore is the dual still $\frac{\partial}{\partial f\left( x\right)}$? I would assume the answer is yes.

Next in stochastic integration, the integral is with respect to Brownian motion, $B\left(t, x,\omega\right)$, usually written $B_{t}\left(x,\omega\right)$.

Brownian motion is a process that doesn't have bounded variation, but instead has quadratic variation $t$, so the integral needs a special definition, which is typically written $\displaystyle \int f\left(x\right) dB_{t}\left(x\right)$. Note the $d$ is with respect to time. Is it appropriate to call $dB_{t}\left(x,\omega\right)$ a "one form"? And is there a notation for taking the one form d of one variable and holding the other one fixed in the case of functions of multiple variables. For example: If I write $d_{t}B\left(t, x,\omega\right)$ vs. $\partial_{t}B\left(t, x,\omega\right)$ I've never seen $d_{t}$, but I'm not sure that the partial notation gives the feel of it being a "one form" even though it would seem to be more correct to me?

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