By Frances Yoon

Some international stock indexes rose, following U.S. markets higher on optimism over signs of an economic recovery and plans for additional stimulus, while shares fell in Hong Kong and mainland China.

In Thursday afternoon trading in Hong Kong, Australia's S&P/ASX 200 had jumped about 1.2%, while Japan's Nikkei 225 added 1.9%.

China's Shanghai Composite and South Korea's Kospi Composite each declined by less than 1%, while Hong Kong's Hang Seng fell 1.8%. The Chinese yuan weakened against the dollar. E-mini S&P 500 futures, flipping between small gains and losses, were up 0.2%.

U.S. stocks surged Wednesday, reaching highs not seen since early March, as the Dow Jones Industrial Average rose 2.2%.

"We have a FOMO rally--a fear of missing out," said Michael Drummey, head of U.S. equity risk trading at Mizuho Americas LLC. "People are frustrated that they missed out on the rally in the past few days, and that frustration is only growing."

Mr. Drummey said investors across the globe are picking up stocks that were sold during the height of the pandemic, but continue to debate whether to buy overvalued stocks, or to invest in companies that still face challenges from a slow recovery.

However, he warned that stocks could be due for a reasonably sized pullback because of the economic uncertainties ahead. "The market is acting in a way that doesn't really line up with that uncertainty," he said.

On Wednesday, U.S. Secretary of State Mike Pompeo said the State Department had determined Hong Kong no longer has a high degree of autonomy from China. That clears the way for President Trump to implement a range of possible measures, including revoking special arrangements on trade.

"Investors are worried about whether that means there could be new trade barriers introduced," said Chang Wei Liang, a macro strategist at DBS Bank. "We're not likely to get a solution on this immediately, so this will be lingering on investors' minds until we get clarity on what the U.S. intends to do with Hong Kong."

Mr. Chang added that the weakness in China's currency also reflected the heightened U.S.-China tensions.

In the offshore markets, the yuan weakened slightly to trade at 7.1802 to the dollar, according to FactSet. That put it close to its weakest levels since China started allowing offshore trading of the currency in 2010. Last September, the yuan depreciated beyond 7.19.

The People's Bank of China set a daily midpoint for trading of the more tightly controlled onshore yuan at 7.1277 to the dollar. That was only slightly stronger than Tuesday's fixing, which was the weakest since February 2008. The onshore yuan was trading at 7.1659 by early afternoon Shanghai time.

Paul Sandhu, head of multiasset quant solutions for Asia-Pacific at BNP Paribas Asset Management, said while trade tensions help explain the weakness in the yuan, Chinese investors' pursuit of higher returns overseas is another reason pressure is building on the currency.

Yields on the 10-year U.S. Treasury note rose 0.02 percentage point to 0.693%. Bond yields rise as prices fall.

U.S. crude-oil prices fell 3.2% to $31.77 a barrel.

Write to Frances Yoon at frances.yoon@wsj.com