Anyone with expensive/rare gear needs to insure it at appraised value. You are going to get burned if it's stolen and it's only insured at whatever replacement value the company deems, and besides which, if you actually get paid for the claim you can't say the instrument is yours anymore, it's the insurance company's. Some people have done that, taken insurance money and then recovered the instrument, but that's illegal as well, and you're better off negotiating with the insurance company.
I have a number of cameras/lenses and the descriptions and serials are on an inventory list with photos of the equipment (cannot mark a camera, it devalues them). All tucked into a safe deposit box with descriptions of other valuables. I have a copy here for quick reference, but it's better to have two, one in a safe place, in case of fires or other destructive acts.
Music equipment shouldn't be indelibly marked for ID if you think you might sell it (there's good ways of marking secretly that can be removed when sold, many already mentioned), just in case it's stolen from a subsequent owner, recovered and then returned to you. You may not know the owner if it's many years later. If it's already marked and can't be removed, you should make out a detailed receipt that describes the markings so the new owner can deal with the markings however they see fit or use it to prove their ownership.