Functions that are Riemann-Stieljes integrable wrt Brownian motion

by Iden   Last Updated September 15, 2017 11:19 AM

Majority of sources discussing stochastic integral (Ito), begins with a discussion why Riemann-Stieltjes integral fails when integrating with respect to Brownian motion. More precisely, they state that it fails because of unbounded variation of Brownian motion (which makes sense intuitively).

In a book of Robert P. Dobrow, the author defines the Riemann-Stieltjes integral with respect to Brownian motion and gives only technical conditions on the integrand function, but not the Brownian motion.

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It seems somehow contradictory to what most of other books say because they usually only mention the unbounded variation of BM.

I actually found another source where this is briefly discussed and it says that bounded variation of an integrator (e.g. BM) is not a necessary condition actually for Riemann-Stieltjes integral to exist.

So is bounded variation really a stronger assumption than necessary for Riemann-Stieltjes integral exist and there are some functions that you can still integrate in a Riemann-Stieltjes sense wrt to Brownian motion?

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