IMO DL does not have a problem with stroller theft. The only thing I've really heard about is that souvenirs have been taken when left in plain site, and this is rare.
For our trip, I made a laminated sign and hung it from the handles. It read, "This Carriage Belongs to the Mead Family" and had a picture of Mickey and the Gang and our phone# on the reverse side. I also wrote our name and phone# in sharpie on the side of our Maclaran in case the sign came off.
We kept our cooler and a few souvenirs in the pocket underneath. However, the souvenirs were always in the bag and usually not easily visible. I never left our camera or anything, but mainly because it was always with me to take pictures.
The only real issue you may have is a CM moving your stroller to aid in traffic flow. So if you park it somwhere and come out of a ride and find it missing, just look around the the entrance/exit and general area of the ride and you should find it.
So, YES! Take your good, expensive stroller that you love!! IMO, you will NOT have any problems. Keep things out of plain site and you'll be fine. You don't need to resort to disgusting "keep away" tactics...I mean, c'mon! Do you really want to push a stroller around with a fake dirty diaper attached? How embarassing!
As far as renting one, I'm not sure on the price. But I do know if you have a Disney credit card, you can rent one per day for FREE! However, I would caution you against doing this....it's like a sea of these things out there! It's difficult to try and find your rented stroller when so many folks have one and sometimes another family might accidently take your rented one without really looking to make sure it's theirs. Plus, I've heard they aren't really comfy and don't work for infants or older kids. They are a jogging stroller style, and my son was 3 at the time and he would never have fit comfortably in one of these!