The 1999 Staff Report: Do insects sleep? - The Straight Dope
Don’t know why the first four paragraphs are in italics; usually only the question and name of the person who asked it are in ital.
Off to take a nap with my pet stinkbug…
From Google: “Most insects are either active only during the day or only at night. When they’re not active, they rest. This state of rest in insects is called torpor, and it’s not exactly like sleep as we know it. During torpor, insects remain very still and don’t respond much to stimuli around them.”
What if he “cuts one” while he’s lying next to your face on the pillow? There’s a reason they call them “stink bugs”, ya know. 
“…even larks and katydids are supposed, by some, to dream.”
I’ve trained him not to do that.