Girsanov's Theorem and Sign Conventions

Why the Sign Convention Matters

A change of probability measure alters which continuous process is Brownian, even though the underlying sample paths have not changed. The sign is fixed by the density: with an exponential density whose stochastic-integral term is negative, the Brownian motion under the new measure is the original Brownian motion plus the integrated kernel. Reversing either sign without changing the other gives an incorrect drift adjustment.

Fix \(T>0\) and a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) satisfying the usual conditions, carrying a Brownian motion \(B_t\) with respect to this filtration. Let \(\theta_t\) be progressively measurable and satisfy \(\mathbb{E}\int_0^T\theta_s^2\,ds<\infty\). All integrals with respect to \(B\) below are Ito integrals; their integrands are progressively measurable and square-integrable on the stated horizon. In particular, \(\theta\) belongs to \(\mathcal{L}^2\) on \([0,T]\). For an integral with respect to a Brownian motion under a changed measure, the corresponding integrand must likewise be progressively measurable and square-integrable under that measure; local versions can be used after stopping when only pathwise local integrability is available.

Novikov’s criterion, a standard result established in the earlier material on exponential martingales, says that \(Z_t\) is a true \(\mathbb{P}\)-martingale and \(\mathbb{E}[Z_T]=1\). Thus \(\mathbb{Q}\) is a probability measure. The exponential is strictly positive whenever it is defined, so \(Z_T>0\) almost surely; consequently \(\mathbb{Q}\) and \(\mathbb{P}\) are equivalent on \(\mathcal{F}_T\). This distinction matters: \(B\) is Brownian under \(\mathbb{P}\), whereas the theorem identifies a generally different process as Brownian under \(\mathbb{Q}\).

The One-Dimensional Girsanov Theorem

The sign can be read directly from the pair of formulas:

The theorem is the standard one-dimensional Girsanov theorem. Its proof by martingale characterization is given in the proof of Girsanov’s theorem; here the point is to state the hypotheses and make the associated signs unambiguous. The density is a true martingale because Novikov’s condition is assumed, not merely a local martingale. Under \(\mathbb{Q}\), \(B^{\mathbb{Q}}\) is a true martingale: Brownian increments are integrable and have conditional mean zero, so \(\mathbb{E}^{\mathbb{Q}}[B_t^{\mathbb{Q}}\mid\mathcal{F}_s]=B_s^{\mathbb{Q}}\) for \(s\le t\).

The converse sign convention follows by applying the theorem to the kernel \(-\theta\). If \(\exp\left(\int_0^t\theta_s\,dB_s-\tfrac12\int_0^t\theta_s^2\,ds\right)\) defines the density, then the transformed Brownian motion is \(B_t-\int_0^t\theta_s\,ds\). The sign in the density’s stochastic-integral term and the sign in the Brownian adjustment change together.

How an SDE’s Drift Changes

The drift coefficient is the coefficient of \(dt\); the diffusion coefficient is the coefficient of \(dB_t\). Suppose an adapted continuous process \(X_t\) satisfies, under \(\mathbb{P}\), \(dX_t=a_t\,dt+c_t\,dB_t\). Here \(a_t\) and \(c_t\) are progressively measurable, and the integrals are well-defined on \([0,T]\) (for example, \(c\in\mathcal{L}^2\) under \(\mathbb{P}\), with the required local integrability under \(\mathbb{Q}\)). Substitution of \(dB_t=dB_t^{\mathbb{Q}}-\theta_tdt\) gives the drift adjustment.

Proof. By definition, \(B_t^{\mathbb{Q}}=B_t+\int_0^t\theta_s\,ds\), so \(B_t=B_t^{\mathbb{Q}}-\int_0^t\theta_s\,ds\). Substituting this identity into the integral form of the SDE gives

The substitution is valid first for bounded, simple integrands by linearity of the sums defining the integrals. For progressively measurable integrands for which the displayed integrals exist, it extends by the defining limits of stochastic integration, or by localization when the integrals are only locally defined. The finite-variation term is the ordinary time integral \(\int c_s\theta_s\,ds\). This proves the stated drift and diffusion coefficients. \(\square\)

This algebra does not assert that every chosen \(\theta\) is admissible: the density and measure change require the hypotheses of the theorem. Nor does changing measure change the pathwise coefficient \(c_t\) in this representation. It changes the drift term when the original Brownian motion is expressed using the Brownian motion under \(\mathbb{Q}\).

Worked Sign Checks

Common Mistakes and a Compact Reference

A frequent error is to use the sign from one convention in the density and the sign from another convention in the transformed Brownian motion. For example, with density \(\exp(-\int\theta\,dB-\tfrac12\int\theta^2ds)\), writing \(B^{\mathbb{Q}}=B-\int\theta ds\) is incorrect. It fails the defining relation \(B=B^{\mathbb{Q}}-\int\theta ds\) used to rewrite the original SDE. The correction is to use \(B^{\mathbb{Q}}=B+\int\theta ds\), and then \(dB=dB^{\mathbb{Q}}-\theta dt\).

Also distinguish a true martingale from a local martingale. Novikov’s condition is a sufficient condition ensuring that the exponential density is a true martingale with expectation one; without a condition that guarantees this, a positive stochastic exponential may be only a local martingale and may fail to have expectation one. In that case it cannot simply be used as a probability density. The theorem’s hypotheses are therefore part of the sign convention’s application, not optional technical decoration.

The table uses \(\mathbb{Q}\) for the measure defined by the indicated density, provided the relevant Novikov condition holds. Its second row follows from applying the first convention with kernel \(-\theta\). The general check is mechanical: identify the kernel from the density, write \(B\) in terms of the new Brownian motion, and only then substitute into the SDE.