Girsanov theorem and drift changes

Let $\theta$ be predictable with a finite squared time integral almost surely on the horizon. The exponential with negative stochastic integral and negative half of its squared integral is a positive local martingale. To define a probability measure by its terminal value, it must have expectation one. A strict local martingale can have smaller expectation, so nonnegativity and a formally canceled drift do not by themselves supply a probability density. This is the central integrability requirement of the theorem.

Novikov's condition, finiteness of the expected exponential of half the accumulated squared shift, is a convenient sufficient criterion for the exponential to be a true martingale. It is not necessary in full generality. A bounded shift on a finite horizon satisfies it immediately. The assertion is horizon-specific: true-martingale densities for every finite time need not produce a uniformly integrable density on the infinite horizon or an equivalent measure on the full infinite-time path sigma algebra.

Define the new measure on the terminal sigma algebra by density $Z_T$ relative to the original measure. Because the exponential is strictly positive almost surely, the two finite-horizon measures are equivalent. Under the new measure, the corrected process $W_t^Q=W_t+\int_0^t\theta_sds$ is Brownian for the filtration on that horizon. With the opposite sign in the stochastic exponential, the Brownian correction would have the opposite sign as well.

Substitute $dW=dW^Q-\theta dt$ into the original SDE. Its new drift is $b-\sigma\theta$, while the diffusion coefficient is unchanged. In several dimensions the same calculation uses a vector shift multiplied by the noise matrix. A drift change outside that matrix's range cannot be produced by such a shift. Pathwise quadratic variation is unchanged by adding finite variation or by passing between equivalent measures, even though the distribution and expected trajectory can change.

Equivalence preserves null events on the chosen horizon, not numerical probabilities or expected values. A process with one mean under the original measure can have another mean under the new one. Likewise, a drifted coordinate process need not become standard Brownian unless its new drift is zero and its covariance and initial value have the standard form. A volatility of two still has variance growth four after drift removal, so rescaling remains necessary.

When a desired shift is unbounded, stopping can verify local versions, but a global measure change still requires a mean-one terminal density or another justified criterion. One must not silently remove the stopping times from a density argument just because an SDE solution itself is nonexplosive. Integrability of the likelihood and nonexplosion of the state are distinct properties. Keeping them separate is particularly important when measure changes are used as a weak-existence method rather than merely to simplify a solved equation.