Game Of Thrones: Is Tyrion Lannister Actually A Targaryen?
Why Would Tywin Raise Tyrion?
If Tywin did indeed know that Tyrion was a bastard of Aerys, as opposed to being his true-born son, then why wouldn't he have him killed, or at least shipped him off to some far-flung corner of the continent?
Above all else he is a man obsessed with the honour and status of his House. While it may damage his inner-pride to raise a child that isn't is - one who is a dwarf, no less - to simply cast him aside would surely raise more questions, and to admit that his wife was raped or had an affair and had a son because of it would perhaps, in Tywin's eyes, be much more damaging to his reputation.
There's also the fact that he dearly loved Joanna. We hear talk of how he was a different person when she was alive, and that he even smiled at their wedding - and Lord Tywin never smiled. Whether Tyrion is his son or not, he is definitely Joanna's, and he may have raised him because of that.
There is also the possibility of Tywin not knowing about it, in which case he would have no reason to not raise Tyrion, although this would be less likely as not an awful lot slips past Tywin Lannister, even if he chooses to turn a blind eye to it.