I recently started taking a medication which raises my blood pressure and I understand very little about how high blood pressure affects the body. I do know that high blood pressure can lead to a stroke though.

I happen to exercise a lot which I know raises my blood pressure, is it possible that I could actually have a stroke from exercising?

Lots of things can lead to stroke. Surely since you've been prescribed a medication which would raise your blood pressure your doctor has made sure you were healthy enough for it first and asked about other risk factors/family history.

How much of a BP increase are you seeing and how long is it staying up high?