* European shares on course for 10th day of gains * Asian shares buck trend and fall on China jitters * Oil prices drop again after demand warning LONDON, Aug 13 (Reuters) - European stocks hit new highs on Friday and were on course for a record-breaking run, capping another strong week as investors seize on a dip in U.S. inflation and more forecast-beating corporate earnings. It was a different story in Asia, where worries about a regulatory crackdown in China and a surge in the COVID-19 Delta variant has sapped confidence. U.S. inflation numbers this week suggested rising price growth may be peaking, which would ease pressure on the Federal Reserve to begin tapering its asset purchases. Pandemic-era stimulus has been behind much of the surge in stock prices the past year, but a stronger than expected economic rebound in much of the world and massive corporate earnings has given the rally new legs in recent weeks. By 0810 GMT on Friday, the MSCI world equity index , which tracks shares in 50 countries, was just below an all-time record high. The broader Euro STOXX 600 was 0.15% higher - on Thursday it equalled its longest ever longest winning streak. Friday would see the index extending gains for a record tenth consecutive session. Markets in Germany and France added 0.2%. Britain’s FTSE 100 gained 0.3%. Futures also pointed to a small gain on Wall Street when it opens. Not everyone is convinced the recent rally can continue, however. “We feel a bit more cautious headed into autumn because of uncertainty on the health front, the Chinese regulatory front and the monetary policy front,” said Paul O’Connor, head of multi-asset at Janus Henderson, stressing he was not “bearish by any means” but had dialled back exposure to riskier assets. “The perception is we have passed the high point in terms of central bank generosity that has suppressed yields. We should expect higher nominal and real yields from here which should start to chip away at the highest valued parts of the markets -U.S. and tech,” he added. In Asia, markets mostly declined. MSCI’s broadest index of Asia-Pacific shares outside Japan fell 0.65%, and was 0.87% lower for the week. Chinese blue chips weakened 0.55%, dragged down by its local semiconductor sub-index, which slumped 4.1%. More broadly, “rising regulatory and geopolitical risks are weighing on medium-term growth prospects (in China), especially in segments targeted by national reform or security effort,” private bank UBP said in an note. The dollar held firm on Friday, staying near its highest level in four months against a basket of currencies, as investors looked for more hints from the U.S. Federal Reserve on its plans to reduce monetary stimulus. The euro edged higher but at $1.1739 was not far off four-month lows. Nearly two-thirds of economists here polled by Reuters said the Fed is likely to announce a taper of its asset purchases - currently set at $80 billion of Treasuries and $40 billion of mortgage-backed securities per month - at its September meeting. The yield on benchmark 10-year Treasury notes was last down 2 basis points at 1.3472%, against a U.S. close of 1.367%. Oil prices fell for a second straight day after the International Energy Agency warned that demand growth for crude and its products had slowed sharply. Additional reporting by Sujata Rao in London and Alun John in Tokyo Editing by Mark Heinrich Our Standards: The Thomson Reuters Trust Principles.