# Moments of a SDE: a detail on the information set

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.

• 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. – Kevin Dec 7 '20 at 8:34
• @Kevin Thanks for the precision. That's what I was suspecting. – Stéphane Dec 7 '20 at 16:01