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Note that SDE (4) does have a "closed-form" representation. Let $X$ be $$X := S^p, $$ so (4) is a geometric Brownian motion SDE $$dX = (p\alpha + 2^{-1}p(p-1) \sigma^2) X dt + p \sigma X dW, $$ which, again due to Ito Lemma, is equivalent to $$ d \ln X = (p\alpha + 2^{-1}p(p-1) \sigma^2 - 2^{-1}p^2 \sigma^2) dt + p \sigma dW $$ or $$ d \ln X = ...


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