* Banks, miners lead Europe shares higher; Asia holds steady
* Dollar stronger versus rivals as Treasury yield rise
* Oil up on U.S. drilling fall but OPEC output rise limits gains
* Euro zone manufacturing activity highest since April 2011
* Graphic: World FX rates in 2017 http://tmsnrt.rs/2egbfVh
LONDON, July 3 (Reuters) - European shares began the new quarter with solid gains and Asian shares held near two-year highs, while the dollar lifted from nine-month lows as U.S. Treasury yields hit their highest since mid-May.
Oil prices rose as U.S. drilling activity declined for the first time since January, though gains were capped after a Reuters survey showing OPEC output was its highest of the year last month.
The pan-European STOXX 600 index, which suffered its biggest month loss in a year in June on worries over tightening monetary conditions, rose 0.7 percent in early trade, led higher by banks and basic resources firms.
France's CAC 40 index rose 0.8 percent, Spain's IBEX 0.9 percent and Italy's FTSE MIB 1 percent. Britain's main FTSE 100 index added 0.3 percent.
MSCI's broadest index of Asia-Pacific shares outside Japan held steady, staying within a stone's throw of a two-year peak hit last week.
Japan's Nikkei ticked up 0.1 percent, buoyed by a Bank of Japan survey showing confidence among big manufacturers hit its highest in more than three months in June.
Chinese blue-chip shares dipped on worries the world's second-biggest economy could be slowing down.
Hong Kong's Hang Seng index rose 0.1 percent, with financial shares benefiting from the launch on Monday of the "Bond Connect" scheme linking China's $9 trillion bond market with overseas investors.
The dollar index, which measures the greenback against a basket of currencies, rose 0.2 percent as U.S. Treasury yields rose. Ten-year yields hit a high of 2.33 percent and last stood at 2.31 percent, up 1.6 basis points.
The euro, which hit 14-month highs against the dollar last week after European Central Bank President Mario Draghi hinted at tweaks to the bank's bond-buying stimulus programme, fell 0.3 percent to $1.1394.
In anticipation of growing economic strength, manufacturing activity in the euro zone, as measured by IHS Market's purchasing managers' index for Junes, hit its highest since April 2011.
The yen fell 0.4 percent to 112.81 to the dollar after Japanese Prime Minister Shinzo Abe's Liberal Democratic Party lost an election in Tokyo on Sunday, potentially signalling trouble ahead for the premier.
Sterling slipped 0.1 percent to $1.3005. In another hint of tighter policy to come, Bank of England Governor Mark Carney said last week a rise in interest rates was likely to be needed as the economy came closer to running at full capacity.
However, some analysts are not persuaded the BoE will hike as Britain negotiates its exit from the European Union.
"What may have been a subtle attempt by BoE officials to put the question of policy normalisation on to the table has in fact led to markets now pricing in odds of a 2017 rate hike equivalent to a coin toss," ING currency strategist Viraj Patel wrote in a note.
BUND YIELDS PULL BACK
German 10-year government bond yields, the benchmark for euro zone borrowing costs, pulled back from 3 1/2-month highs hit in the wake of Draghi's comments.
It last stood at 0.47 percent, down 0.3 bps.
"The market reaction last week was very sharp and now is probably the time to digest recent comments," said Patrick Jacq, European rates strategist at BNP Paribas. "July could be a more supportive month for bonds as there is less supply, but clearly the trend is now upwards for yields."
Brent crude oil rose 22 cents to $48.99 a barrel after a survey on Friday showed U.S, drillers cut the number of rigs in use by two to 756 last week. The total was still more than double the number a year ago.
Gains were limited, however, as a Reuters survey on Friday show OPEC oil output rose by 280,000 barrels a day to a 2017 high, despite the group having agreed to cut production to help balance the market.
"To put that in context, that is nearly a quarter of the 1.2 million barrels (per day) OPEC agreed to cut," said Greg McKenna, chief market strategist at AxiTrader, adding this increase was driven by higher output from Nigeria and Libya, which were exempted from the cuts.
For Reuters Live Markets blog on European and UK stock markets see reuters://realtime/verb=Open/url=http://emea1.apps.cp.extranet.thomsonreuters.bi z / c m s / ? p a g e I d = l i v e m a r k e t s
(Additional reporting by Hideyuki Sano in TOKYO, Jemima Kelly and Dhara Ranasinghe in LONDON; Editing by Raissa Kasolowsky)