Can you slow the shutter speed with a digital camera?
You want a long exposure, sometimes called bulb mode. Most compacts do not give you this option. Usually only the top end compacts have manual controls for aperture and shutter speed. if you can't find the info in your manual, it's probably not an option. All dSLRs can do this however.
The technique you are referring to is called light painting.
Can you slow the shutter speed with a digital camera?
With that model of cybershot?
I think so.
Oh, wait.
You didn't tell us what model you were using.
I don't know.
You come on here asking for help but don't provide basic information.
Sony make and have made a lot of cameras.
General hint: if it has a dial with the letter 'S' on it then you probably can.
More general hint: we are a helpful bunch on here, Just give us the basics.
If it's like my PowerShot, then you set the dial to M and you can control all the settings manually. But it probably won't work in the video mode.