Strollers getting stolen at WDW are rare if at all. Now, CM will move them from time to time, since guests often park them in areas they shouldn't. The bigger concern is someone mistakingly thinking yours is theirs since a lot of people have the same stroller, so I agree with the above posters that some kind of 'identifer' will work wonders. Either that or rent one from the parks, or just buy a cheap umbrella stroller before your trip that you can afford to lose. When I was down there with my daughter last year, I left our nice stroller at home and just made do with a $20 umbrella from Babies R Us. It broke on the last day, which was fine because my intent was to trash it anyway.