You can do this very easly. You cable company may have a SOHO package for another $30 USD/mo. Mine does, COX.NET. With the SOHO package (or whatever they call the home buniess package) you can get a static IP address so you run your server. A static IP is about $5/mo.
If you don't want to use or pay for a static IP address the other option is Dynamic DNS. This is a service that allows you to use a Dynamic IP address to host a server. There are many companies out there but I have used to and would recommend them
The first is NO IP.com
. It has a free element as well as premium feature and hosting services.
The other is Zone Edit