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Comment by seanhunter | Hacker News Reader

Not exactly an answer to your question but also from finance. A fun moment after the 2008 financial crisis came when a lot of pundits were trying to call the bounce. Everyone kept going on MSNBC and the like saying that “the second derivative was positive” and saying that was a sign the economy was about to turn around. I was watching one day and yet another person said that for some indicator (maybe nonfarm payrolls?) the second derivative was positive and therefore things were about to get better when Mohammed El-Erian (who was also on the program) said dryly “There are lots of analytic functions where the second derivative is consistently positive and the first derivative is consistently negative”. There was a stunned silence.

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As a simple example, one can look at f(x)=1/x in R+. The derivative is f'(x)=-1/x^2 and the second derivative is f''(x)=2/x^3. In general, every function of the form f(x)=1/x^n with odd n will have this behaviour, and if n is even the behaviour is the opposite. And yes, many more examples can be found.

More generally, this is why I'm wary of indirect indicators. They never tell the whole story, and because of that they're used disingenuously in order to muddy the waters. You see this a lot in big scale PR campaings, such as climate change denialism, and pro-sugar and pro-alcohol disinformation (we had a lot of pro-tobacco as well, but it has subsided in the last two decades or so, at least in the West).

Yes exactly. There are folks who would ride that positive second derivative all the way down to f(x) = 0 and believe every step of the way that the indirect indicator is telling them that we're going to turn around any second now.

For those who find the existence of such analytic functions intuitively "wrong", note that it is essential to this example that though the second derivative is positive, it is positive and decreasing towards 0. If the second derivative is bounded from below by some positive value, then eventually the first derivative will become positive and it will diverge towards positive infinity and thus the function itself will diverge towards positive infinity.

More precisely suppose that f is twice continuously differentiable and f'(x) < 0 and f''(x) ≥ 0 for all x greater than some K (for example K=0 in the examples given) then λ( (f'')^-1((a, ∞)) ∩ (K, ∞) ) < ∞ for all a > 0 where λ is the Lebesque measure.

A good example for pundit disproving is f(t) = exp(-t), the solution to the differential equation dx/dt = -x. So the continuous limit of something like "every year, 1% of the ice caps melt." Look, the volume of ice cap lost is decreasing year over year!

Also, while we might accept a certain smoothness hypothesis (fuzzy logic as applies to human perceptions) in financial valuations, these functions certainly aren't (if I may be pedantic) analytic at all.

If a function is analytic, then the derivatives at one point tell the whole story on the complex plane--the information is all encoded at (an arbitrarily small neighborhood around) a single point. But most smooth functions aren't analytic; indeed, a smooth function that is the trace of a process taking any stochastic external input will typically (probability 1) not be.

Yes, and El-Erian certainly understands that. He was using classes of analytic functions as a relatively simple example of things that had the properties he was talking about.

Generally speaking in finance things like price trajectories and timeseries like economic indicators are modelled as stochastic processes. Often a Wiener process with drift, and then jumps and jumps in volatility added as needed to capture the dynamics of the situation if required. So there's definitely no requirement to be smooth or diferentiable everywhere, and there is even less requirement for a positive 2nd derivative to lead to a positive slope/turnaround.

Hi @seanhunter, any chance you have a source for this?

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