Stocks pushed broadly higher on Tuesday, ending a volatile trading session in which the market fluctuated widely between gains and losses.

The surge by the end of the day came after the market showed signs of recovery from the worst sell-off on Wall Street in more than two years, even as it wavered throughout the trading session.

The three main indexes, for example, gyrated wildly in the half-hour after a Federal Reserve announcement that included no new steps to pare interest rates further or to foster economic growth, shedding gains, plunging sharply and recovering their losses.

At the close, the Dow Jones industrial average was up 429.92 points, or nearly 4 percent, at 11,239.77. Standard & Poor’s 500-stock index was up 53.07 points, or 4.74 percent, at 1,172.53, the largest point gain in a day since March 2009, and the Nasdaq was up 5.29 percent, or 124.83 points, at 2,482.52. The yield on the benchmark United States Treasury 10-year notes fell to almost 2.2 percent, from 2.32 percent late Monday. Two-year yields, already at a record low, were down to 0.186 percent.

It was a marked contrast to the close on Monday when the markets had spiraled downward on the first trading day after Standard & Poor’s downgraded its rating of the United States government’s long-term debt.

In the statement, the Federal Reserve said it would hold short-term interest rates near zero through mid-2013, but it announced no new measures.

Dan Greenhaus, the chief global strategist for BTIG LLC, noted that the Fed acknowledged in its statement that risks of further deterioration have increased and that it specified how low rates would be and for how long. Altogether, Mr. Greenhaus said, “the implication is clearly that the Fed is moving to a more accommodative policy, albeit one that doesn’t do much to boost economic activity.”

Investors had been awaiting the meeting of the Fed’s policy board for any guidance or signals about the economy or the possibility of the Fed injecting any further monetary stimulus.

Benchmark crude declined 2.5 percent to close at $79.30 a barrel.

Investors have been hoping that the recent turmoil in global stock markets would abate, but they expected indexes to remain unsettled a day after Wall Street’s worst showing since December 2008.

Douglas Coté, the chief market strategist for ING Investment Management said in remarks before the Fed announcement that he did not believe a new program of stimulus would be introduced, and that policies to cut spending and promote growth were needed.

“It is just going to delay the true fixes that need to take place,” he said.

The downgrade worsened the already low sentiment in the marketplace, battered by sovereign fiscal problems in Europe and worries over the economy in the United States.

Peter V. Coleman, head of equity research at JMP Securities, said that although investors seemed focused on the downgrade, its effect was mostly psychological. Otherwise there was little relevance to the financial markets, he said, pointing to the fact that United States government bonds were still in demand on Monday even after the rating was lowered.

“The downgrade did not really have any relevance,” he said.

“Obviously, we are getting a bit of a relief trade,” Mr. Coleman said.

If the gains are sustained through the end of the day, it will be a welcome sign of relief for investors, although analysts do not expect a smooth ride ahead.

“We can expect a rocky road,” Mr. Coté said. “What happened over the weekend was a ratcheting up of global risk. A downgrade of the global reserve currency creates increased risk.”

“It might have been a little overdone,” he added, noting that corporate profits in the United States could provide some ballast.

“It’s going to be an up and down market,” he added. “It’s like water, it finds its right level.”

The aftermath of the downgrade and Wall Street slump continued to reverberate in global markets as well.

Stocks in Asia closed mostly lower for the day and European stocks fell sharply before gaining ground before the close. The Euro Stoxx 50 index, a barometer of euro zone blue chips, closed up 0.3 percent, while the FTSE 100 index in London ended the trading day up 1.9 percent. Prices for energy companies like Royal Dutch Shell and industrial giants like Siemens, businesses that stand to suffer in an economic downturn, fell sharply early in the session, but were in positive territory by the afternoon.

The dollar declined against other major currencies. The euro rose to $1.4235 from $1.4179 late Monday in New York, while the British pound fell to $1.6202 from $1.6318. The dollar fell to 77.12 yen from 77.76 yen and to 0.7288 Swiss francs from 0.7550 francs.

Investors continued to seek safe-haven assets. Comex gold futures rose 1.8 percent to $1,740 an ounce, having reached above $1,770.

Adding to the risks faced by investors is the potential for the upheaval to weaken the wider economy by constraining the ability and willingness of banks to extend credit to businesses and making it harder for companies to raise money from the capital markets.

The fear among investors has reached epidemic proportions, with the sell-off erasing $8.1 trillion — or 14.8 percent of market capitalization — from global stock markets since July 24.

Asian trading saw enormous volatility, with steep early declines partially reversed as the day progressed. But the Kospi index in Seoul closed 3.6 percent lower and the Tokyo benchmark Nikkei 225 stock average fell 1.7 percent. In Hong Kong, the Hang Seng index fell 5.7 percent and in Shanghai the composite index closed essentially flat.

Alone among the big markets, the Sydney benchmark S&P/ASX 200 index closed Tuesday in positive territory, with a 1.2 percent gain.

The European Central Bank continued its purchases of government debt, news agencies reported. That helped to send Spanish and Italian 10-year bond yields lower for a second day, with Spain below 5 percent for the first time this year, and Italy at 5.05 percent.

The bank’s president, Jean-Claude Trichet, urged Spain and Italy to quickly shore up their credibility with the markets, and called on European leaders to move quickly to carry out measures agreed on July 21 to bolster the euro zone’s bailout fund.

“Taken together, and particularly since Lehman Brothers, this is the most grave crisis we have faced since World War II,” he said in an interview with Europe 1 radio in Paris, citing the “financial turbulence” that started in August 2007. “And I believe that it would be the worst since World War 1 if the authorities had not taken the important decisions they have.”