When you stream to a DVR stream, your content will stay on that stream until it's replaced or manually cleared. Most DVR streams are set up to hold up to 2 hours of content. If you never clear your stream, then as you stream more, the new content will push out the old, only holding 2 hours at a time. This also means that after you're finished live streaming, your viewers will be able to rewind and watch the last two hours of your stream.
If you would like to keep your DVR clear, and don't want viewers to be able to watch the stream after you've finished, you can clear the content yourself. To do this, you'll go to your Live Monitor and select your stream. Scroll down to the Actions button underneath the preview window, and select Reset DVR.