The Sensex at the Bombay Stock Exchange rose 12 points, or 0.1 percent, to close at 19,441, after volatile trade, as the rupee hit a record low against the dollar on Monday. But the Nifty at the National Stock Exchange declined 3 points, to 5,878.Stock markets in Japan, Hong Kong, South Korea and Singapore gained between 0.2 percent and 4.9 percent.