The altitude is wrong across the entire DEM? Is the offset consistent across the DEM? For example, if you know the area is 10m higher across the entire DEM, then you could add 10 to each pixel value using something like Band Math.
http://www.harrisgeospati...m/docs/BandMath.html

However, if you have GPS points showing a few specific pixel locations , you would only have enough information to correct those pixel locations only. There is not enough information to interpolate the elevations of other pixels across the DEM. If you have many GPS points across the area and wanted to triangulate those points to make a new surface, you could use Rasterize point data.

http://www.harrisgeospati...rizingPointData.html