Legal Question in Landlord & Tenant Law in Missouri

The first place to look to see who is responsible for ridding the property of pests is the written lease. Often they tell each party which will handle extermination costs. Generally, the Landlord is responsible for providing the property free of pests, and the tenant is responsible for keeping it that way. Since you say the roaches appeared several months after you moved in, so you probably are the source, or they got in on your watch.. They could have entered through an open door/window, with groceries or with people or pets that have entered the premises. Check your lease, If that doesn't resolve you question, talk with your landlord, to see if you can come to some joint effort to rid the property of roaches. If you allow the infestation to continue with out informing the landlord, you may relieve them of their duty to do anything about it during your tenancy.

Good luck