AN F900 AND A B777 HAD COORDINATED TCASII RA'S IN C90 CLASS B AIRSPACE.

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FALCON 900 DEPARTING MDW RESTR TO 3000 FT BY MDW TWR. B777 ARRIVING INTO ORD; ON BASE LEG AT 4000 FT FOR ILS RWY 22L APCH. FALCON 900; WHILE STILL ON MDW TWR FREQ; SAID HE WAS RECEIVING TCASII ALERT; TOLD TO CLB ABOVE 3000 FT BY RA. CLBED UP TO 4100 FT; THEN DSNDED BACK DOWN TO 3000 FT. IT WAS AT THIS POINT I FIRST TALKED TO FA900. VERIFIED ALT. ASKED HIM A FEW QUESTIONS; CLAIMED HE GOT A TCASII ALERT VERY SHORTLY AFTER TKOF. B777 PLT APPARENTLY SAW FA900 POP UP THROUGH LOW LEVEL CLOUD/FOG LAYER. I BELIEVE HE ALSO GOT RA FROM TCASII BUT FROM FA900'S CLB. FA900 BUSTED THROUGH 3000 FT TO AVERT COLLISION WITH WHO? THAT IS THE MILLION DOLLAR QUESTION. ALL I CAN THINK OF IS THAT THE FA900 GOT A TCASII ALERT FROM A PREVIOUS ORD ARR. EVEN THOUGH THAT PREVIOUS ORD ARR WOULD BE WELL OFF HIS R; NO OTHER TFC WOULD HAVE BEEN IN THE AREA DUE TO WX. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR HAD A CHANCE TO REVIEW THE 'RAPTOR' ATC RADAR TAPE. HE STATES THAT THERE WAS NO THIRD ACFT PRESENT. HE ALSO STATES THAT RAPTOR DOES NOT SHOW ALL RADAR TFC PRESENT. THE ORIGINAL FALCON TCASII RA OCCURRED IN MDW AIRSPACE SO THIS CTLR WAS NOT INVOLVED UNTIL THE FALCON WAS HANDED OFF TO HIM. SUPPLEMENTAL INFO FROM ACN 500825: ON DEP FROM MDW GOT A TA TFC. ADVISED TWR AND WAS TOLD NO TFC IN THE AREA. CONTINUED CLB FOR 3000 FT ASSIGNED. TA WENT TO RA WITH CLB COMMAND. ADVISED TWR AND DEP CTL WE WERE STARTING A TCASII CLB. CLBED ABOVE ASSIGNED ALT. WHEN CLR OF CONFLICT; RETURNED TO 3000 FT.