Forget about the "quietly" for a while, and concentrate on the "relaxed".
The key to this is where the power behind the playing is coming from. Imagine you are playing one of the Three Witches in Macbeth, stirring a huge steaming cauldron to brew up your next spell. How would you do that? Most likely, you would be moving your whole arm, or even your whole upper body, as you stirred. You need to play these octaves the same way, but obviously with the size of the movements scaled down. The muscles in your shoulder and upper body have enough power and stamina to do this for hours without getting fatigued, so use them!
Moving your wrist and fingers is just to "pick off the right notes" as your whole arm moves "in a circle" for each group of four octaves. If the power is coming from your wrist, fatigue will set in before you get to the end of the passage. Actually, it's possible that as your wrist gets fatigued and "locks up", you do (unconsciously) start to use your arm muscles more, from necessity - which is why the notes get louder as the fatigue sets in.
Practice this "pot-stirring" movement on an imaginary giant-sized piano (i.e. using a table as an imaginary keyboard), before you scale it down to the size of the real instrument.