Is there any way to own a direct connection to Amazon Web Services?

A.
No, AWS only allows access from the public Internet.

B.
No, you can create an encrypted tunnel to VPC, but you cannot own the connection.

C.
Yes, you can via Amazon Dedicated Connection.

D.
Yes, you can via AWS Direct Connect.

Explanation:
AWS Direct Connect links your internal network to an AWS Direct Connect location over a
standard 1 gigabit or 10 gigabit Ethernet fiber-optic cable. One end of the cable is connected to
your router, the other to an AWS Direct Connect router. With this connection in place, you can
create virtual interfaces directly to the AWS cloud (for example, to Amazon Elastic Compute
Cloud (Amazon EC2) and Amazon Simple Storage Service (Amazon S3)) and to Amazon Virtual
Private Cloud (Amazon VPC), bypassing Internet service providers in your network path.
http://docs.aws.amazon.com/directconnect/latest/UserGuide/Welcome.html