If it's not a silly question, when you say "cut it down (to a stump)", just how brutal are you?
The first time I tried overwintering I did that and killed the plant off; the second time I didn't prune at all and it survived - but maybe it wasn't the best way to manage the plant.
This year I am trying to overwinter 2 jalapeno varieties and want to try to get it right as, not having a greenhouse or the best weather in the UK (we are in Yorkshire), a head start next year would be nice.
Best regards,
Sue