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Edit:I am not 100% sure that what I suggested is correct (though I swear I saw something like this in a class)... but now I would like to make sure I get the correct understanding in my mind.

My understanding at the moment is:

To make proper sense of a Reimann-Stieltjes integral of the form \int f dg, you need that one of f and g be continuous and the other be of bounded variation. Which is which doesn't matter because you can define the other from the first via integration by parts.

Since W_t is continuous a.s., you can then define \int h(t) dW_t omega-wise a.s. provided h(t) is of bounded variation. This way of defining "stochastic integration" fails however for \int W_t dW_t since neither of the pieces is of bounded variation... hence the need for more advanced notions of stochastic integration.

However, I believe that the different notions of stochastic integration coincide when the integrand is of bounded variation. And so, provided your integrand was of bounded variation, you could think in terms of RS-integration. And therefore the integration-by-parts I suggested would be legitimate.

Zhoraster's observation does raise some concern (although it could be that the sum of two non-abs continuous functions is abs continuous) so now I am curious if my mental picture is wrong.

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I believe you can use integration by parts to express \int X_s ds as -\int s dX_s + Boundary terms.

This is then a stochastic integral of the type commonly dealt with.