The Wiener process rings a bell here ))) which text book are you using?
To answer your question, I'd say if std variation is 1, the variance is also 1, so it gives b in either case.
Before I state my question, I'll quote my textbook on financial derivatives:
I'm not very good at math, so this is why I'm asking this question. I thought it wasn't possible to multiply st. dev., only variance. Then why is the st. dev. of a generalized Wiener process b times 1.0? Thanks[...] A generalized Wiener process for a variable x can be defined in terms of dz as
dx = a dt + b dz (12.3)
[...] The b dz term on the right-hand side of equation (12.3) can be regarded as adding noise or variability to the path followed by x. The amount of this noise or variability is b times a Wiener process. A Wiener process has standard deviation of 1.0. It follows that b times a Wiener process has a standard deviation of b.