The generalization of this is a martingale. https://en.wikipedia.org/wiki/Martingale_(probability_theory) … Your expected value at the next time step is the same as your present value, but your variance grows over time as you keep reinvesting your winnings. In the long run, you get rich...or go broke.
What I’m wondering about specifically is an asset whose value is exp(sigma W_t), where W_t is the Wiener process and sigma is a constant. That is, geometric Brownian motion with no drift.
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I initially thought this would have expected value equal to 1, but double-checked Ito calculus & apparently it’s actually exp(sigma^2 t), if I didn’t make a calculation mistake? Would an agent maximizing log-return invest anything in this asset?
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Wikipedia has it at exp(mu*t). https://en.m.wikipedia.org/wiki/Geometric_Brownian_motion#Properties …
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yeah, no, that confused me too the first time, that's after a change of variables. the wiki page there says the prob distribution that has mean exp(mu*t) is exp((mu - sigma^2/2)t + sigma W_t). The drift term is *not* the same as the exponent in the mean.
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