The term in NJ is "Notice to Quit and Vacate" for a rent increase and change of terms in a lease. For example, you can state NO new or replacement pets allow, but you can NOT demand a dog remove if it is on the lease or that the landlord has allowed to be living at the residence for 90 or more days. Now if the dog has biten someone, you can demand immediate removal of "that" dog from the property. If you do not issue a new lease, the terms of the expiring lease remain in effect.
But as for raising the rent 10% every month, it becomes "unconsciousable" after the first raise within a 12 month period. If this is a HUD lease, your rent increase maximum is determined by ther regional FMV (fair market value) of rents as established by HUD. You are NOT allowed to charge "side payments" or to demand that they move. If you fail to perform repairs as stated on the annual inspection, you CAN NOT evict them for failure to pay rent. Instead, you are abated (flush down the toilet) all rents til you make and have inspected the required repairs. If you have Rent Control in the town/city/boro/muncipality, you might be limited to a much lower rent increase. A good legal aid lawyer might be able to reduce that older 10% level to 2 or 3% in the current economic rent values.
So you can demand ANY rent you so dream of, tenant does not pay, you try to evict, you lose due to UNCONSCIOUSABLE rent increase, you pay the LAWYERS fees for tenant, judge might give you $2 per month or can lower the rent, and you just learned what it is to be a landlord in NJ.
You might hear the term, landlord declined to renew my lease --- trust me, there is at least 3+ notices for not complying with a standard lease. Damages, police calls, illegal activities, over-occupancy, habitually late payment of rent, failure to perform terms in the lease (trash, not mowing grass, pets). They got the letter to QUIT and VACATE by a certain date.
NJ was an original colony; tenants would clear land, build shelter, and install fences, while owners where the original land grants from the King of England. They were NOT serfs and hence, the common and written law evolve with far more protections for the tenant to have a home and not to be run off by the property owner. Large grant land owners soon found it better to just sell the ground.