there is no edge tool.
You set the extent of the raster slightly larger than the raster. Assuming that the raster is rectangular and axis-oriented, find the current extent, then set the Left, right, top and bottom appropriately so you have a border of nodata values.
Numpy has specify tools for 'padding' arrays (aka, rasters) and functions that perform moving/rolling functions that account for nodata.
I suspect most of esri's functions account for edge effects behind the scenes, so before spending a lot of time on this, compare a simple neighborhood mean with a 3x3 moving window with and without the nodata collar.