Living in an apartment like this is difficult, at best. Lots of people have used a form of "Portable Panels" that have helped. For info here's a quote:
"Portable panels: If you plan on moving but still want some immediate relief from sound annoyances, make some portable panels you can take with you. Buy 4'X8' wall paneling (also called "Panel-board". "Homasote" and ordinary "Soundboard" also works well.) from the hardware store and attach Super Soundproofing Mat to the backside with contact cement. Use at least 1" thick material. (PN 09-42730). (Well, OK, 1/2" may work pretty good, too). Push the panels up into place on your wall using a long finishing nail here and there to hold them in place. (Tip: seal the edges of your panels with our acoustical caulk for best results). A noticeable sound level reduction will be made and you can take your panels with you! For even more reduction, buy low-cost acoustical ceiling tiles and attach them to your walls first. You can leave them behind because they are so inexpensive. (So is a material called "Wonderboard" Also Duroc, which is made of concrete!). Caulking around the edges of your panels as mentioned above is essential."
And here's a link to info on the SSP Mat (see first product discussed)
http://www.soundproofing.org/sales/prices.html You will still have the structure-born noise problem caused by all the running and juming children. This is because structure-born noise is caused by impact events or mechanical vibrations. The acoustic energy of these vibrations is transmitted through solid structures (floors. walls, cielings, studs, joists, pipes, etc.) and then released as air-born sound somewhere else in the building. It can only be directly solved using some form of de-coupling at the source.