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And in practice if the initial conditions are not explicitly specified, what Mathematica does, for example, is to use as an initial condition a representation of various features of the exact state of the computer system at the time when Random was first called.
But one does not imagine that this means that the planet itself contains a device that explicitly solves such equations.
But experts in the particular types of systems involved have usually been quick to point out that there are many details that my models do not correctly reproduce.
But it is my strong suspicion that such purposes in fact have very little to do with the real reasons that these particular features exist.
But I do not believe that this is in fact the case.
So how does this work?
So how does this work?
And in fact, to do so requires absolutely no sophisticated ideas from mathematics or elsewhere: all it takes is an understanding of how to apply simple rules repeatedly.
Practical experience suggests that particularly on short timescales much of the randomness that one sees is purely a consequence of internal dynamics in the market, and has little if anything to do with the nature or value of what is being traded.
But the point is that to do so would require a computation far more sophisticated than any that could meaningfully be done as part of a practical measurement process.