(Corrects grammar in headline)

* Graphic: 2020 asset performance http://tmsnrt.rs/2yaDPgn

* Graphic: World FX rates in 2020 http://tmsnrt.rs/2egbfVh

MILAN, Oct 5 (Reuters) - Stocks and other risk assets rose on Monday as signs that Donald Trump's health was improving brought relief to markets after the uncertainty of his COVID-19 infection sent investors rushing for safety last week.

The U.S. President, 74, was flown to a hospital for treatment on Friday, but his doctors said he had responded well and could return to the White House as soon as on Monday.

The MSCI world equity index, which tracks shares in 49 countries, was up 0.4% by 0812 GMT, supported by overnight gains across Asia and a positive start in Europe.

The pan-European STOXX 600 rose 0.7%. S&P 500 futures rose 0.5% and Nasdaq futures 0.8%, indicating a similarly strong start on Wall Street later.

Overhanging the relief rally, however, were concerns that Trump's case could be more severe than public disclosures suggest, and that more restrictive measures by governments to slow coronavirus infections could harm the economic recovery.

Some traders were concerned by doctors' admission that Trump had been given supplementary oxygen and steroids.

"Many questions remain including the use of the steroid drug ... which is usually reserved for those with severe illness," said Raymond James strategist Chris Bailey in London. "Global cases now top 35 million and various new restrictions in Paris, New York, etc",

A survey on Monday showed the euro zone's economic recovery faltered last month as new restrictions sent its dominant service sector into reverse.

IHS Markit's final composite Purchasing Managers' Index fell to 50.4, just above the 50 mark separating growth from contraction.

Trump's infection also comes less than one month before the presidential election on Nov. 3, potentially fuelling more market volatility and making the outcome of the vote even more difficult to predict.

"In terms of the impact on the election, we haven’t seen enough polling to assess whether this increases or decreases his chances of winning," said Deutsche Bank strategists.

According to a Reuters/Ipsos poll released on Sunday, Democrat contender Joe Biden opened his widest lead in a month in the U.S. presidential race.

The volatility VIX index, known as Wall Street's fear gauge, remained close to the one-week high it hit on Friday.

Meantime, suggestions Trump could leave hospital sent oil prices up more than 2%. An escalating workers' strike in Norway that has shut four of Equinor's oil and gas fields also helped drive the gains.

Brent prices were up 2% at $40.1 a barrel and U.S. West Texas Intermediate added 2.2% to $37.9 a barrel.

The dollar was little changed as investors awaited news about U.S. Trump's health and developments in fiscal aid talks in Washington. The dollar index was last down less than 0.1% on the day at 93.722.

Yields on benchmark 10-year Treasuries rose to 0.7088% and the yield curve steepened slightly, signalling investors felt comfortable taking on more risk.

But euro zone bond yields edged lower on concerns about possible new restrictions to fight the coronavirus.

The French government has announced new restrictions, closing bars for two weeks. Other countries across Europe are also weighing up more measures.

Gold, which is often bought during times of uncertainty, fell 0.35% to $1,892.1, highlighting increased risk appetite.

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(Reporting by Danilo Masoni in Milan, additional reporting by Stanley White in Tokyo; editing by Larry King)