The laws vary by state, county, and city. Presumably, your Realtor knows what the regulations are in Agoura Hills.
There's nothing you can do to speed up the law. The process is usually very specifically defined: You issue your "pay or quit" notice. Often it's a 3-day or 5-day notice, but again that'll vary. Then, if the tenant hasn't paid or left, you then can go to the court and get an order that'll lead to the sheriff coming out. But you can't (and shouldn't) cut any corners. Ten days is actually super-quick.
However, if the deed hasn't transferred yet, that's really the responsibility of the seller. First, at this point it's the seller who has standing with the court. You're just an interested party. The owner is the one in the position to evict. The tenant doesn't have a lease with you.
Further--and your Realtor can advise you, as can a lawyer--you shouldn't close on the property with a squatter in there.
There is one way, maybe, to speed up the process: Cash for keys. You pay the squatter to move. You don't pay until after the squatter has moved out, though. But you make an offer: $500, maybe. Or $500 and moving expenses. Or $1,000. Use your own judgment. Sometimes that's the quickest and easiest way to get a deadbeat out.
Again, check with your Realtor and with a lawyer.
Hope that helps.