Can I bank start a fraud investigation without you signing for it to be done?

Why wouldn't you want a fraud investigation?  Let's hear the rest of the story.

Of course they can.  

Of course they can.   If they suspect you are committing fraud, they dont need your permission to investigate

You might ASK them to do a fraud investigation if you think someone is cheating you, but they do investigations all the time simply because they suspect someone is up to no good. That someone might be you, a customer.

Of course any bank can investigate fraud, they do it daily, they have their own fraud investigation department

No, because they would just need to prove it.

Yes, in a word.            .

Yes, you don't need my permission.

That sounds like asking if the police have you get your permission before investigating you for committing a crime.