If SESSION_START_DATE_TIME is of type TIMESTAMP you may want to try using the SQL function TO_TIMESTAMP. Here is an example:
SQL> CREATE TABLE t (ts TIMESTAMP);
Table created.
SQL> INSERT INTO t
2 VALUES (
3 TO_TIMESTAMP (
4 '1/12/2012 5:03:27.221008 PM'
5 ,'mm/dd/yyyy HH:MI:SS.FF AM'
6 )
7 );
1 row created.
SQL> SELECT *
2 FROM t
3 WHERE ts =
4 TO_TIMESTAMP (
5 '1/12/2012 5:03:27.221008 PM'
6 ,'mm/dd/yyyy HH:MI:SS.FF AM'
7 );
TS
-------------------------------------------------
12-JAN-12 05.03.27.221008 PM