When people call that DID, I dont want to take the call, instead I want to divert it to an 800 number I own, lets say 888.888.8888, using the FROM_DID as the calling party.
111.111.1111 calls my pbx at 555.555.5555
The call is immediately diverted to 888.888.8888 looking like its coming from 555.555.5555
111.111.1111 is connected to 888.888.8888 to complete the call transaction, but 888.888.8888 behaves as though the call has come from 555.555.5555
If you do it with your Asterisk system, extra costs may occur. Technically it may depend on what your provider accepts from your side.
I am doing something like this with a twist as I want to keep all the CDR records in my local PBX. In that case I simply register others phones like mobile phones to Asterisk. Depending on the contract this may also cause extra costs.
Use the Transfer application without answering the call. This assumes support by the provider. If they support it, they will, normally, bill it as though you had dialled it as an outgoing leg.