The MSCI's broadest index of Asia-Pacific shares outside Japan was up 0.1% in early trade.
Australian shares were 0.2% higher while South Korean shares opened nearly flat, pressured by tensions in the Korean peninsula.
North Korea suspended its sole remaining major project with the South on Monday, the Kaesong industrial park, amid speculation that it will take some sort of provocative action - another nuclear weapons test or missile launch - in what has become one of the most serious crises on the peninsula since the end of the Korean War in 1953.
Alcoa Inc , the largest US aluminium producer, reported an increase in quarterly profit on Monday, easing concerns about corporate results in the first three months of 2013.
Earnings forecasts have been scaled back heading into the first-quarter reports, with S&P 500 company earnings seen up just 1.6% from a year ago, according to Thomson Reuters data, down from a 4.3% forecast in January.
Japan's Nikkei stock average opened up 0.9%, after jumping as much as 3.1% to its highest since August 2008 earlier on Monday.
Early in Asia on Tuesday, the dollar hit its highest since May 2009 of 99.65 yen while the euro climbed as far as 129.72 yen, its highest since January 2010. The Aussie dollar soared to 103.78 yen, the highest since July 2008.
Traders expect the dollar to trade at 100 yen, as soon as this week, as the effects of the BOJ's latest move continue to spread globally.
"Markets are increasingly focused on the notion that larger JGB purchases at longer maturities by the BOJ could push Japanese domestic long-term investors elsewhere," said Vassili Serebriakov, strategist at BNP Paribas.
The BOJ conducted its first government bond buying operation on Monday since announcing monetary stimulus on an unprecedented scale last week, saying it will buy 1.2 trillion yen of Japanese government bonds.
A sharp decline in Japanese government bond yields across the curve has stirred speculation Japanese investors will turn to higher-yielding assets in a global scale, pushing sovereign bond yields lower in the United States and the euro zone.
Concerns over the euro zone were heightened when Portugal's constitutional court rejected some of the austerity measures introduced as a condition of the country's bailout, but a jump in euro zone bond yields were contained by sharp falls in yields of Spain and Italy due to demand for higher-yielding euro zone bonds from Asia after the BOJ plan.
"The global market reaction to the BOJ stimulus has been more positive for duration in fixed income than equities so far," said Barclays Capital in a research. "We think markets are not giving the BOJ enough credit for its support. We continue to favour equities."
US crude futures were up 0.1% to $93.46 a barrel.
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