Starting to record in the middle of a live streaming program

Rokus have no provisions for recording anything.  Some premium channel apps include DVR services where programs can be recorded on the channels' own servers, but how you access the DVR controls is up to the particular channel's programmers. 

They could place them jn the Roku's normal Options popup accessed via the Roku remote * key, or they may choose other ways to access them.  One commonly used alternative is to access them via screen overlays that pop up when the up and/or down cursor keys are pressed.