Good question, Emina - a couple important notes from an agent with experience in this matter:
1) Your apartment is under RENT STABILIZATION if the building you live in was built before 1978 and it has a Los Angeles city address. That applies to the vast majority of apartments in Los Angeles, and in that case, you are entitled to stay in your apartment and pay the same rent (with a 3% yearly increase).
2) Sometimes when the bank owns the property, they don't bother collecting rent. That doesn't mean you don't OWE it! The new owner is entitled to collect all of the rent you haven't paid, so save your money and hope the new landlord isn't savvy.
If you or any landlords have questions about rent control, contact me at my blog below.