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How to find expectation of this stochastic process? Also, to show that the expectation of a stochastic process expression [Xt - St] in one measure is equal to expectation of another expression (of the mentioned stochastic process) in another measure?

Given: $S_t=S_0e^{\sigma W_t+(r-\sigma^2/2)t}$

$dS_t = rS_tdt + \sigma S_tdW_t$

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  • $\begingroup$ What is your dynamic of $S_t$? $\endgroup$ – nakajuice Mar 27 at 3:23
  • $\begingroup$ $S_t=S_0e^{\sigma W_t+(r-\sigma^2/2)t}$ It took me forever to type this up using latex. $\endgroup$ – happyGiraffe Mar 27 at 4:09
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    $\begingroup$ Hint: take the log of $S_t$, write out the integral in the exponent and you'll have one non-random term and one term which is $\sigma \int_0^t W_u \mathrm{d}u$. Then see quant.stackexchange.com/questions/29504. This integral is normally distributed which should help you compute the expectation you are looking for. $\endgroup$ – LocalVolatility Mar 27 at 8:34
  • $\begingroup$ Please show your work. What have you tried, and how far have you gotten? Also, if this is a homework assignment, it should be marked as such. $\endgroup$ – AdB Mar 28 at 9:57

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