JSESSIONID cookie is created/sent when session is created. Session is created when your code calls request.getSession() or request.getSession(true) for the first time. If you just want to get the session, but not create it if it doesn't exist, use request.getSession(false) -- this will return you a session or null. In this case, new session is not created, and JSESSIONID cookie is not sent. (This also means that session isn't necessarily created on first request... you and your code are in control when the session is created)
Update: Every call to JSP page implicitly creates a new session if there is no session yet. This can be turned off with the session='false' page directive, in which case session variable is not available on JSP page at all.