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Given a portfolio of shares $I = \sum{w_iS_i}$ for some fixed weights $w_i$ where each stok $S_i$ has a log-normal distribution, what is the process / distribution followed by the portfolio? That is, what is the distribution of the process

$${dI} = \sum{w_idS_i} = \sum{w_i(\mu_iS_idt+\sigma_iS_idW_i)} ?$$

Given that the log-normal distribution is not closed under the summation, it would imply that the process followed by the portfolio is NOT log-normal. However, it seems that it is, nevertheless, still modeled as log-normal in applications. Why is this so? And how bad/good this approximation is to the "true" process given above?

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