Can somebody briefly introduce the mathematical aspects, in particular, those related to mathematical finance, of the three economists who were just awarded this year's [Nobel Memorial Prize in Economic Sciences](https://en.wikipedia.org/wiki/Nobel_Memorial_Prize_in_Economic_Sciences)?


According to the [New York Times](https://nyti.ms/GUETdc):



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> The three economists, who worked independently, were described as collectively illuminating the workings of financial markets by showing that stock and bond prices move unpredictably in the short term but with greater predictability over longer periods.
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This seems very related to mathematical finance: all kinds of random noises, yet in the long run, may not deviate too much from, say, their rational means. Many thanks for the comments, introductions and ideas!



On mathematical aspects of the most recent Nobel Prize in economics winners' work
Just to add a little something to arsmath's very good answer: The mathematics in Fama's main idea that returns are impredictable are indeed simple, and moreover, not due to him. What Fama did is a huge empirical study to support that claim. For the mathematical argument itself, which is simple but not absolutely trivial, in other words which has some mathematical content, 
I recommend any interested reader this very nice article of Samuelson (1965), [Proof that properly anticipated prices fluctuate randomly](http://www.ifa.com/Media/Images/PDF%20files/Samuelson-Proof.pdf).


