= Girsanov theorem
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= Cameron-Martin-Girsanov theorem
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The Girsanov theorem describes how an absolutely continuous <change of measure> alters the <drift> of a <semimartingale>. In a Brownian version, let $\theta$ be <predictable> with $\int_0^T\theta_s^2ds<\infty$ <almost surely>, and suppose
$$
Z_t=\exp\!\left(-\int_0^t\theta_s\,dW_s-\frac12\int_0^t\theta_s^2ds\right)
$$
is a true <martingale> on $[0,T]$. Its initial value is one. Under $dQ=Z_TdP$, the process $W_t^Q=W_t+\int_0^t\theta_sds$ is standard <Brownian motion> through time $T$. <Novikov's condition> is one sufficient test for the density assumption.<Applied Stochastic Differential Equations by Simo Särkkä and Arno Solin>[Section 7.3, Brownian change of measure; the sign here corresponds to a negative stochastic-exponential integrand]{ref}
For $dX_t=\mu dt+\sigma dW_t$ with constant $\sigma>0$, choosing $\theta=\mu/\sigma$ gives $dX_t=\sigma dW_t^Q$: the same paths have different statistical weights and zero <drift> under $Q$. Their <quadratic variation> is unchanged. A positive <local martingale> density is insufficient unless its <expectation> remains one, and consistent finite-horizon changes need not give an equivalent infinite-horizon measure.
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