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Also, while we might accept a certain smoothness hypothesis (fuzzy logic as applies to human perceptions) in financial valuations, these functions certainly are
by psi75 4y ago
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.
- seanhunter 4y agoYes, 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.