When I visited Seoul, we went all over the city and I realized after 4 days that I had not seen a single policemen nor heard a single police siren. Seoul is 25 million people. I live in a "safe" California town of 60,000 people and see policemen all the time, and hear sirens regularly. I would caution any friend from jogging at 2am here.
In coffee shops in Seoul, people leave their wallets and laptops unattended while they go to the bathroom. By contrast, in my hometown in California, thieves have walked in, punched customers, grabbed laptops out of the customer's hands and run.
A very different world -- not just Seoul, but other cities in Asia.
In my time, at 2am UK streets were full of drunk men and women. Is it safer to be drunk than practicing sport? Did it change so much in 10 years or is it a safety fantasm?
Except that a woman jogging in Seoul's street on 2am would be a totally believable sight: I wouldn't personally recommend it, but it wouldn't look too out of place.