I'd be curious what a mathematician thinks but I've seen things like the PDE I used in my Kolmogorov posts referred to as an analytic solution sometimes contrasted against simulation which is described an alternative "brute force" approach. I'm not so sure anymore. In watching the software do the finite differences approximation for the PDE I see it do this: it creates a little fake world in a matrix with 200 increments of time in one dimension and 5000 increments of wealth units in another. Then it does 5000 calculations by passing one direction through the wealth dimension and 5000 calculations going another direction and then repeats that another 199 times over the time dimension. Maybe that's not "simulation" but it's an awful lot of work in a fake world to come up with a guesstimate that will become rapidly unreliable shortly...sounds an awful lot like simulation to me, just by another name.
The same, I think, could be said for PWR (sometimes called the maximum withdrawal rate) where it's a dead analytic equation that assumes it has a known sequence of returns. Cool enough, but it doesn't really come to life until you breath volatility into it through simulation.
They (analytic and simulation) are all destined to be failed crystal balls, though, because they are trying to make an end run on the future which is, in the end, unknowable.
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