Fill in missing dates with data value from previous populated date for group
Use LEAD() to get the next row within the TicketId partition. Then join to a Calendar table to get all the dates between.

WITH TAwithnext AS
(SELECT *, LEAD(AssignmentDate) OVER (PARTITION BY TicketID ORDER BY AssignmentDate) AS NextAssignmentDate
 FROM TicketAssignment
)
SELECT t.TicketID, c.Date, t.DepartmentID
FROM dbo.Calendar c
JOIN TAwithnext t
    ON c.Date BETWEEN t.AssignmentDate AND ISNULL(DATEADD(day,-1,t.NextAssignmentDate),t.AssignmentDate)
;


All kinds of ways to get a Calendar table...