Is there a way to correct or compensate for camera shake in the video?
I have. There are multiple ways to do this, but the most common and effective methods are:
Select the clip you want to correct on the timeline and open "Stabilization".
Click the red circle for "Stabilization" to enable it, then click the "Stabilization" name to expand the menu.
Press <Stabilize> to analyze the clip and reflect the results.
You can change the correction settings on this screen. We recommend that you try a few and use the better settings.
The following site explains the options in detail. For your reference.