inheriting a tenant with no written agreement - options?

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Hi friends,

I'm closing on a property soon where I'm inheriting a tenant who's been living there for over a year.  There's no written agreement or lease; the tenant is paying about half of market rent currently (I think--I don't even know the exact amount he's paying because there's no agreement).

Laws here require giving tenants 90 days for a no-cause eviction or to raise rents more than 5%.  Do I just have to wait the 90 days, or is there any way to shorten that time period, especially since there's no agreement? I could move into the house myself if needed.  (Law says the seller can give just 30 days notice if the buyer is moving into the property herself, but that notice must be given within 120 days of contract signing, and it's been longer than that.)

The plan is to have him sign an agreement on day 1, so it's not a question of whether to have an agreement or not.  It's more about what I can legally do--if anything, aside from cash for keys--to bring the rent closer to market faster, given that there's no written agreement currently, and the verbal agreement is with the seller, not with me.