I have a question that might be trivial to most of you, but somehow I'm not able to solve it by myself. I have a disagreement with my colleague on the distributional properties of a Geometric Brownian Motion: my point of view is, that if you estimate the parameters mu$\mu$ and sigma$\sigma$ on log return (and assume that they are normal), the GBM at point t$t$ has indeed an expected Value of X0exp((mu+sigma^2/2)t) (properties of log-normal) and not X0exp(mut$X_0\exp{((\mu+\sigma^2/2)t)}$ (properties of log-normal) and not $X_0 \exp{(\mu t)}$ as found in the literature, since there mu$\mu$ is the drift term of the differential equation of the stock price process itself, not of its log-returns. I I tried to confirm my view via several derivations and numerical examples.My My colleague though is still not convinced, cause I use dlnX$\text{d }{\ln{\!X}}$ in Ito's Lemma for the log returns, but he argues that they are ln(Xt/Xt-1)$\ln{(X_t/X_{t-1})}$. Apparently, my knowledge of differential equations is too limited to find the step from the log-returns to their corresponding sdeSDE. In the literature I only find derivations where they start with the sde of X$X$. I'm looking foward to oyuryour hints.
