How do I know if the calculations of my (adjustable) mortgage payments have been correctly made through the years?

My wife and I have an ARM (adjustable rate mortgage) with only a relatively small balance remaining, so we’re thinking of paying it all off. But this got me to thinking… All these years, we have simply paid whatever amount the lender’s servicer said was due each month. I suppose most people who have ARMs just do the same.
Now, the interest rate on an ARM can change periodically, and there are other parameters involved such as the index, the margin, the adjustment period, the ceiling (i.e., that maximum interest rate), and the floor (the minimum interest rate). In addition, this mortgage has been sold many times, and even the servicer is different from the one we started with. Not only that, but the index is the 11th District Cost of Funds, one that is not very popular any more.
With so many variables, I can see many opportunities for errors. Is there any way to check whether the payments have been calculated correctly through the years?

If there is anyone who has worked in the real estate, banking or mortgage industries, I’d appreciate any information you could give me or direct me to. Thanks!