Logarithm of geometric Brownian motion

First identity: apply [F1] to $g(t,y)=x\exp ((\mu -{\sigma }^2\mathrm{/}2)t+\sigma y)$ along $\widehat{B}$, which is indistinguishable from $B$ and has drift $0$ and diffusion $1$. Then ${\mathrm{\partial }}_{t}g=(\mu -{\sigma }^2\mathrm{/}2)g$, ${\mathrm{\partial }}_{y}g=\sigma g$ and ${\mathrm{\partial }}_{yy}^{2}g={\sigma }^{2}g$, so $d{X}_t=\mu X_{t}dt+\sigma X_{t}d{B}_t$ up to indistinguishability. Positivity and continuity hold everywhere by the explicit definition. The composition defining $X$ is adapted, so [F1] also makes $X$, $\mu X$ and $\sigma X$ progressive and predictable. If $C_T$ is the upper bound in [F2], then ${\int }_0^T\mathrm{\mid }\mu X_s\mathrm{\mid }ds\le \mathrm{\mid }\mu \mathrm{\mid }T{C}_T$ and ${\int }_0^T(\sigma X_s{)}^{2}ds\le {\sigma }^{2}T{C}_T^2$ on every path. Together with $X_0=x$ and the displayed decomposition these verify the Ito-class assertion explicitly.

Second identity: because $X$ was defined by a positive exponential, taking the ordinary logarithm gives the everywhere pathwise identity $\log X_t=\log x+(\mu -{\sigma }^2\mathrm{/}2)t+\sigma {\widehat{B}}_t\mathrm{.}$ Since $\widehat{B}$ and $B$ are indistinguishable, this is exactly $d\log X_t=(\mu -{\sigma }^2\mathrm{/}2)dt+\sigma d{B}_t$ up to indistinguishability. The explicit bounds of [F2] also verify directly that $X$ stays in a compact subinterval of $(0,\mathrm{\infty })$ and its logarithm is bounded on every finite horizon; no localization or SDE existence theorem is being smuggled into the argument.

Boundary and consistency cases: at $t=0$ the identities read $X_0=x$ and $\log X_0=\log x$; for $\sigma =0$ the formulas reduce to the deterministic exponential and its logarithm; for $\mu =0$ the drift of $\log X$ is $-{\sigma }^2\mathrm{/}2$; and $x>0$ is required for the logarithm. The formulas concern the explicitly defined process, not existence for an SDE, and AC enters only through [F3].