How to handle Cease and Desist from huge company

Unfortunately your question is way too vague, each case is unique. Also you didn't originally say which country you are in.

There are a couple of facts which should be considered (assuming USA):

1) Do they have a federal trademark? You should get a copy on-like for free from the Patent & Trademark Office. They may be bluffing?

2) Does their trademark pre-date your use of your name? If you started using your name before they started using theirs you will most likely have a prior claim and should be able to continue using your name.

3) Assuming they have a federal trademark, and they had the trademark before you stated using your name, then the question is will it likely cause confusion? There is a Cadillac dog food as well as a Cadillac car; this is fine because nobody is likely to confuse dog food for a car.

4) Finally how big is the other company and how well known? If it is McDonalds, CocaCola or Disney who regularly and actively defend their trademarks you should think twice before going up against them.

If after considering the above you still want to keep the rights to your name you should consult with a intellectual property lawyer who regularly handles trademark cases. This is a very specialized field and outside the scope of practice for the average lawyer.

EDIT:

Based upon your comment that this is a letter from a DJ 30 company and that you are presumably much smaller here are a couple of points that may help.

Big companies send out cease and desist letters all the time, usually with virtually no research or merit. You are at a very early stage of the process. Years ago I got such a letter from a firm retained by Microsoft asking me to stop using the name "WinCron" for a piece of freeware I had written. The only time the name had ever appeared in print was in a directory published by Microsoft itself! I simply called up their in-house council (not the firm that sent the letter), pointed out that it was Microsoft, not me , that had published the name and the problem was solved instantly. If you get a good trademark lawyer he may be able to have a similar conversation with the big company an reach an amicable settlement very quickly.

To give you an example of a real case that was handled in just such a fashion there is a small store in Maine Gimbal & Sons, that got just such a letter from Gimbels, the big New York City department store. This similarity, both are stores and have very similar names is probably much closer than yours. They came to terms; remember that most big companies don't want to be know as the big bad guy that runs around ruining people lives and putting people out of work. They too have an incentive to work this out if they can.