Very basic questions. Let $(z_t)_{t \geq 0}$ be a standard Brownian motion and let $$dS_t = \mu S_t dt + \sigma S_t dz_t.$$ When we write $E\left( S_t \right)$, do we mean $E\left( S_t \big| F_0 \right)$? The hesitation stems from my habit with discrete time processes which, in levels, would not be covariance stationary.
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2$\begingroup$ You’d typically start with $\mathcal F_0=\{\emptyset,\Omega\}$ which means you have no information at all. Only constant functions would be measurable, suggesting that we know today’s value, $S_0$. Every integrable random variable is independent of this trivial sigma algebra, i.e. conditioning on it does not refine the standard expectation. $\endgroup$ – Kevin Dec 7 '20 at 8:34
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$\begingroup$ @Kevin Thanks for the precision. That's what I was suspecting. $\endgroup$ – Stéphane Dec 7 '20 at 16:01
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