Hello Lukas
The behaviour that you are describing is actually done by design. When you configure a DMZ, it is designed to be able to communicate with the “internet at large”, so communication from the OUTSIDE interface towards the DMZ is allowed. Take a look at the Security Zones section of the following lesson:
However, communication from OUTSIDE to INSIDE is not allowed. You can however make an exception to this rule by using an access list, and specifying what kind of traffic you will allow taking place in this direction. More info on how this can be done can be found here:
Now one thing to keep in mind is that if the destination of your ping is the ASA interface itself, then you will get no response to the ping. The ASA is designed not to respond to pings that come from a different subnet. This may be the reason that there is no reason given for the drop. Take a look and let us know…
I hope this has been helpful!
Laz