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juked07 comments on A hypothetical question for investors - Less Wrong
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Even after determining your wealth, your utility function has to take whether-you-are-currently-holding-stocks as an input, because it affects the probability that you incur a transaction cost in future time steps. I think this piece cannot be evaluated without supposing some pdf of future pdfs. I think this is why people are saying the problem is "underspecified".
Even then I think we can make some assumptions though. If you have no prior about the pdf of pdfs, a reasonable best best guess is the pdfs you have seen already.