It's perfectly fine and safe to exercise in cold weather as long as you're dressed appropriately for the cold weather and don't over exert yourself when exercising.
Exercising in cold weather is much harder on your body than exercising in warmer weather because during the cold weather your body will be using more energy to try to keep your body warm.
So just take it easy when exercising in the cold and don't overdo it and if you're feeling tired then stop and take a break and get inside somewhere warm for awhile.
I jog or run for around 20 minutes twice per day during cold weather and I'm okay as long as I take breaks during that 20 minute jog.