I noticed this fairly recently, about a year ago - you English-speaking lads and lasses seem to use the possessive form when referring to a person's actions: let's take this quote from Star Ocean: The Second Story as an example:

"I hope you don't mind MY saying so..."

What's this?
Personally I'm more used to the alternative "I hope you don't mind I'M saying so", but perhaps that's just some Swedish thing?

So, what's the point about using possessive forms like this?
Nobody in this country seems to know about it, but still I hear and read it regularly.
I hope you don't mind my asking you.