If you have a loan for the vehicle, the IRS considers that you own it for vehicle expense purposes.
With an owned vehicle, you can choose to claim either the standard mileage or actual expenses method the first year. If standard is chosen, you can switch methods in future years. If actual is chosen, you must continue to use actual expenses.
With a leased vehicle, you can also choose to claim standard or actual expenses method the first year. However, you cannot change methods in a later year, even if you have a renewed lease agreement. Also, if you choose the standard mileage you cannot claim the lease payment.
To claim the lease payment using the actual expense method you must calculate the business portion. See
IRS Publication 463
"Inclusion Amount" on page 23 for more information about that calculation.
Also, to claim business vehicle expenses the vehicle must be in the name of your self-employed business, your name, or your spouse's name only.