The number of blogs commenting on dollar vs stocks inverse correlation is approaching infinitum. I won’t try to summarize what I found, I just want to place some accents.
First let me post the short-term correlation chart (source):

The correlation went positive after BS bailout last year, when everybody went optimistic that the worst is over. Then it fell sharply into inverse territory.
Here’s more long-term view (source):

And even more (source):

While you are waiting for me jumping into explanations let me stress my strong opinion that those charts are describing an extremely complicated phenomena. I won’t be surprised to learn someone earned PhD in economics on those charts, by correlating them once more with various economic conditions.
First of all I think the explanation of this correlation is different at different times. The simplest observation is that the stronger currency is the natural result of the stronger economy, so is the rising stock market (it was this way during the Kondratieff Autumn). When the currency becomes too strong all foreign assets (especially emerging markets and commodities) seem a bit too cheap and money start to search for better return abroad. That triggers the negative correlation as the market loves the weaker (more competitive for exports) currency and continues to rally.
Then it turns out that that the economy is overheated and starts contracting, the money return home and make the currency a bit stronger again and the stock market corrects down. The correlation is still negative. Then the currency starts falling to reflect the weaker economy and the correlation turns positive again. And then the market decides that is wants to be repriced up in the weaker currency and moves up, turning the correlation negative. And so on, but different each time.
The cycle specialists will explain that the oscillation period is different for the currencies and the stocks, producing natural cycles of correlation, the same way anything oscillates in this world.
At present the currency is signaling that the economic situation is dare and the Feds are simply made money too cheap in hope that asset prices will get repriced up. Of course the fast money are pushing oil well before they start pushing real estate, so I’m not sure how much mileage you can get from the cheap money. To illustrate forces at work let me post this chart:

When you push the string like this even the pigs will fly. Pigs do fly, but not the rest of the economy.
Let see the stock market in real constant dollars (source):

By this chart you should see that the stock market was quite oversold in the real dollars and the dollar decline and stock rally is the most efficient way to return from the oversold condition. However you should not try to make predictions based on that, the stock-dollar correlation is extreme and can resolve itself quite a bit unpredictably.

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