His obit from the NY times

http://query.nytimes.com/mem/archive-free/pdf?_r=1&res=9404E7DF153BE533A25754C1A9649C94619ED7CF

makes no comment about the 4th Missouri.

Another source Philadelphia Jews in the Civil War describes his service quite well including his capture and wounding at Richmond VA.

http://books.google.com/books?id=jtoLAAAAYAAJ&pg=PA531&lpg=PA531&dq=brevet+colonel+asch&source=bl&ots=fbp9URntP-&sig=PxkwECFVrMKqZiELZ5weaPzrgGU&hl=en&ei=Dbe6S_CaOITcM4eauM0H&sa=X&oi=book_result&ct=result&resnum=5&ved=0CBYQ6AEwBA#v=onepage&q=brevet%20colonel%20asch&f=false

The Regimental History of the 1st New Jersey Cav. has a number of entries for Asch

http://books.google.com/books?id=mU7ga67RNUYC&printsec=frontcover&dq=1st+new+jersey+cavalry&source=bl&ots=YITEYHV2d7&sig=iF7aOBBgUKuXnfU0eg7boQZ-6rA&hl=en&ei=Tru6S-bwH4HkNcntoZ8H&sa=X&oi=book_result&ct=result&resnum=6&ved=0CBcQ6AEwBQ#v=onepage&q=asch&f=false

I can find no record of any "4th" unit from Missouri having survived beyong 1864. I suspect Col Asch was a New Jersey man alone.

John R