DA50EX Captain reported a slow loss of cabin pressurization resulted in an immediate descent and diversion.

Narrative

I was acting as the pilot-in-command of a part 91 (corporate) flight from ZZZ to ZZZ1 in a Dassault Falcon 50EX airplane. I was accompanied by my Second-in-Command. I was the Pilot Flying. We were at our cruising altitude at flight level 400 and had been cleared directly to our destination. We were about 45 minutes into our scheduled one hour and 20 minute flight. We had three passengers on board.Our first indication of a problem was when we received a CABIN annunciation on the warning panel along with a 'cabin' aural warning. We immediately began scanning the pressurization indications and controls. The cabin altitude was at 10;000 ft. and rising. All of the pressurization controls were in their normal and correct position. I called for the loss of cabin pressurization checklists and Pilot Flying began running the procedures. At this point the cabin altitude was between 15;000 to 20;000 ft. I called for the donning of our crew oxygen masks and we switched our audio panel for masks; and we turned on the fasten seat belt sign and the no smoking sign in the cabin. Then; I called for Pilot Flying to select 'override' on the oxygen controls in order to drop the passenger oxygen masks. I looked into the passenger cabin and noticed that the passengers were not donning their masks; so I yelled from the cockpit for them to put their masks on. Next; following the checklist; we attempted to regain pressurization control using manual pressurization; however we were unable to stop the cabin altitude from climbing. At this point; I decided to initiate an [immediate] descent. I told Pilot Flying to [advise] ATC and to squwak XXXX on our transponder. I established a descent by bringing the power levers to idle; extending the air brakes; and performed a steep bank to the left and a descent near ZZZ2. Our descent rate was approximately 7;000 ft per minute.Center acknowledged our descent and cleared us to descend to 10;000 ft. and asked our intentions. I informed the Copilot that we would deviate and land at ZZZ3. Pilot Flying asked for vectors to that airport and we were assigned heading 60 degrees for vectors.As we approached 10;000 ft.; I began to slow the descent rate in order to level the airplane at 10;000 ft. As I began to level the airplane; I retracted the air brakes and began slowing the plane to 250 kts. I told Pilot Flying that we could take off our oxygen masks at that altitude and put our headsets back on.Then; Center cleared us to descend to 5;000 ft. and we began a normal descent to that altitude. We were approximately 10 miles east of the airport and called the field in sight. Center then handed us off to Approach control. Pilot Flying; checked in with Approach Control and advised that we had the field in sight. Approach Control then cleared us for a left turn back towards the airport that established us on a final approach to Runway XX. They told us the current weather information at ZZZ3. We noticed that our cabin altitude was at approximately 7;000. ATC cleared us for a visual approach to Runway XX and to contact the Tower. We began descending to pattern altitude. Pilot Flying contacted the Tower and we were cleared to land. With the remaining cabin pressurization; I asked the Copilot to dump the cabin to obtain zero pressurization for landing. We were able to perform a normal landing without further incident on Runway XX at ZZZ3.I believe some factors that may have affected our response was the slow climb of the cabin that made it less noticeable. It wasn't a rapid decompression but a slow climb that didn't cause any obvious pressure changes in the cabin. Also; we did struggle for few seconds in setting the audio panel where we were able to hear ATC over the speaker instead of through our headsets. Otherwise; I believe we followed our memory items and emergency procedures as we have been trained. I must commend my Second-in-Command for her outstanding performance.