Change of Measure and Girsanov's Theorem

As a conditional expectation of a fixed integrable variable, $Z$ is a martingale by the tower property, $\E_\P[Z_t\mid\F_s]=\E_\P[\E_\P[Z_T\mid\F_t]\mid\F_s]=\E_\P[Z_T\mid\F_s]=Z_s$. It is strictly positive, for if $A=\{Z_t=0\}\in\F_t$ then $0=\E_\P[Z_t\,\mathbf 1_A]=\E_\P[Z_T\,\mathbf 1_A]$, which with $Z_T>0$ a.s. forces $\P(A)=0$, so $Z_t>0$ $\P$-a.s. and the division by $Z_s$ below is legitimate. For $A\in\F_t$ the defining property of conditional expectation gives $\E_\P[Z_t\,\mathbf 1_A]=\E_\P[Z_T\,\mathbf 1_A]=\E_\Q[\mathbf 1_A]=\Q(A)$, where the middle equality is the defining change of measure on $\F_T$, valid for $A$ because $A\in\F_t\subseteq\F_T$.

The right side is $\F_s$-measurable, so it suffices to check the averaging identity over an arbitrary $A\in\F_s$. Extending Proposition 1 from indicators to $\F_t$-measurable $Y\ge0$ by monotone convergence, and then to integrable $Y$ by splitting into positive and negative parts, gives $\E_\Q[Y]=\E_\P[Z_t Y]$; here $Y=\mathbf 1_A X$ with $\E_\P[Z_t|X|]=\E_\Q[|X|]<\infty$, so every term is finite. Using this at level $t$, then the tower property,

and the last expression equals $\E_\Q\big[\mathbf 1_A\,\E_\P[Z_t X\mid\F_s]/Z_s\big]$ by Proposition 1 at level $s$. The two sides agree over every $A\in\F_s$, which is Equation (1).

By Levy's characterization it suffices that $\widetilde W$ is a continuous $\Q$-local martingale with $\langle\widetilde W\rangle_t=t$. The drift $\int_0^t\theta_s\,ds$ is continuous of finite variation, so it has zero quadratic variation and contributes nothing to the bracket; hence $\langle\widetilde W\rangle=\langle W\rangle$, which equals $t$ $\P$-a.s. This bracket is a single random variable, computed from $\P$-a.s. limits of squared increments, so the identity $\langle\widetilde W\rangle_t=t$ is a $\P$-a.s. statement; since $\Q\sim\P$ share the same null sets it holds $\Q$-a.s., as Levy under $\Q$ requires. For the martingale property, the product rule applied to $Z_t\widetilde W_t$ gives $d(Z\widetilde W)=Z\,d\widetilde W+\widetilde W\,dZ+d\langle Z,\widetilde W\rangle$. The bracket sees only the local-martingale parts, $Z\theta\,dW$ of $dZ$ and $dW$ of $d\widetilde W$, so $d\langle Z,\widetilde W\rangle=Z\theta\,d\langle W\rangle=Z\theta\,dt$. Substituting this together with $dZ=Z\theta\,dW$ and $d\widetilde W=dW-\theta\,dt$, the drift terms cancel, leaving $d(Z\widetilde W)=Z(1+\theta\widetilde W)\,dW$, a driftless Ito integral whose integrand $Z(1+\theta\widetilde W)$ is continuous and adapted, hence locally bounded, so by localization $Z\widetilde W$ is a continuous $\P$-local martingale. Since $Z$ is a strictly positive $\P$-martingale and $Z\widetilde W$ is a $\P$-local martingale, $\widetilde W$ is a $\Q$-local martingale, because multiplication by $Z$ is a bijection between $\Q$-local martingales and $\P$-local martingales, the local form of the Bayes correspondence of Theorem 2 [2]. That is exactly the $\Q$-local martingale property that Levy's characterization consumes.