I'd read "moving in one direction with Brown motion" to mean Brownian motion plus a linear term. If margin trading causes you to lose the asset, its value goes

I'd read "moving in one direction with Brown motion" to mean Brownian motion plus a linear term. If margin trading causes you to lose the asset, its value goes to zero, hence log of that could be said to be negative infinity. (I'm not endorsing or disputing the conclusion of the comment).