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On the finite horizon $[0,T]$, strict positivity gives equivalence of the restricted probability measures: for $A\in {\mathcal{F}}_T$, $\mathbb{Q}(A)={\mathbb{E}}_{\mathbb{P}}(Z_T{1}_A)$ vanishes if and only if $\mathbb{P}(A)$ vanishes. The standard change-of-measure theorem gives semimartingale invariance under equivalent measures: a process is a semimartingale under $\mathbb{P}$ on this horizon if and only if it is one under $\mathbb{Q}$. The sample-path quadratic variation is unchanged, but the local-martingale and finite-variation parts generally change.

More explicitly, set $N=\int Z^{-1}dZ$, the zero-start stochastic logarithm of the density. For every continuous $\mathbb{P}$-local martingale $L$, the Girsanov theorem states thatis a continuous $\mathbb{Q}$-local martingale. Consequently, if the continuous semimartingale decomposition under $\mathbb{P}$ is $X=X_0+L+V$, its decomposition under $\mathbb{Q}$ isThe covariation correction is a continuous finite-variation process, locally well-defined because $Z$ stays strictly positive. In the Brownian case, if $d{Z}_t=Z_t{\theta }_{t}d{B}_t$, thenis a Brownian motion under $\mathbb{Q}$. The inverse finite-horizon density is $1/Z$, giving the reverse change of measure. These assertions apply on each horizon on which the density is strictly positive; they do not assert equivalence on a larger terminal sigma-algebra where the full density $D$ may vanish.