Time-Dependent Deterministic Drift Changes

From a Deterministic Drift to a New Measure

Changing Brownian motion by a deterministic drift changes its probability law without changing the covariance of its increments. The time dependence matters: the accumulated drift is an integral, so the mean at time \(t\) need not be a constant multiple of \(t\). On a finite time interval, the density and the resulting Gaussian law can be calculated explicitly.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\) with respect to \((\mathcal{F}_t)\). Fix \(T<\infty\). Let \(\theta:[0,T]\to\mathbb{R}\) be a deterministic measurable function with \(\int_0^T\theta(s)^2\,ds<\infty\). The stochastic integrals below are Ito integrals: \(\theta\) is progressively measurable and belongs to \(\mathcal{L}^2\), since \(\mathbb{E}\int_0^T\theta(s)^2\,ds=\int_0^T\theta(s)^2\,ds<\infty\).

The sign convention is fixed by the density: with the plus sign in \(\int\theta\,dB\), the process \(B_t-A(t)\), not \(B_t+A(t)\), will be Brownian under \(\mathbb{Q}\). The measures are equivalent on \(\mathcal{F}_T\), because \(Z_T\) is finite and strictly positive almost surely.

The Density and the Shifted Brownian Motion

Proof. For \(0\le s<t\le T\), the increment \(\int_s^t\theta(u)\,dB_u\) is independent of \(\mathcal{F}_s\) and is normally distributed with mean zero and variance \(v(t)-v(s)\). This follows first for deterministic step integrands from independent Brownian increments, and then for deterministic square-integrable integrands by the Ito isometry and approximation by step functions. The same approximation establishes the stated normal law. The moment-generating function of a centered normal variable with variance \(q\) is \(\exp(q/2)\) at argument \(1\). Therefore the conditional expectation of the density increment is \(\mathbb{E}[\exp(\int_s^t\theta\,dB-\tfrac12(v(t)-v(s)))\mid\mathcal{F}_s]=1\). It follows that \(\mathbb{E}[Z_t\mid\mathcal{F}_s]=Z_s\). In particular, \(\mathbb{E}[Z_t]=\mathbb{E}[Z_0]=1\), and \(Z_t\) is integrable and is a true martingale, not merely a local martingale. This also verifies the density requirement. Novikov’s condition holds here, since \(\mathbb{E}\exp(\tfrac12\int_0^T\theta(s)^2\,ds)=\exp(v(T)/2)<\infty\), though the direct conditional-expectation calculation already proves the martingale claim.

To identify the law under \(\mathbb{Q}\), take any \(0\le s<t\le T\) and real \(u\). The conditional expectation formula for a change of measure gives \(\mathbb{E}^{\mathbb{Q}}[e^{iu(B_t-B_s)}\mid\mathcal{F}_s]=Z_s^{-1}\mathbb{E}[Z_Te^{iu(B_t-B_s)}\mid\mathcal{F}_s]\). The factors in \(Z_T/Z_s\) after time \(t\) have conditional mean one and are independent of the increments up to \(t\). On \([s,t]\), the pair \((B_t-B_s,\int_s^t\theta\,dB)\) is jointly Gaussian, with variances \(t-s\) and \(v(t)-v(s)\), and covariance \(A(t)-A(s)\). The Gaussian characteristic-function formula thus gives \[ \mathbb{E}^{\mathbb{Q}}[e^{iu(B_t-B_s)}\mid\mathcal{F}_s] =\exp\!\left(iu(A(t)-A(s))-\tfrac12u^2(t-s)\right). \] This is the characteristic function of a normal variable with mean \(A(t)-A(s)\) and variance \(t-s\), independent of \(\mathcal{F}_s\). Hence \(B_t-A(t)\) has continuous paths, starts at zero, and has independent centered normal increments with the Brownian variances; it is Brownian under \(\mathbb{Q}\). Since \(B_t=A(t)+(B_t-A(t))\), its mean under \(\mathbb{Q}\) is \(A(t)\). Its covariance is that of the shifted Brownian motion, namely \(\min(s,t)\), because \(A\) is deterministic. \(\square\)

This is the deterministic-integrand case of Girsanov’s theorem. The drift coefficient \(\theta(t)\) is not itself the mean of \(B_t\); the mean is its accumulated integral \(A(t)\). Also, changing measure does not alter the diffusion coefficient here: the centered process under \(\mathbb{Q}\) still has Brownian variance \(t-s\) over an interval \([s,t]\).

Worked Calculations

Finite-Dimensional Gaussian Laws and the Sign Pitfall

For any times \(0\le t_1,\ldots,t_n\le T\), the vector \((B_{t_1},\ldots,B_{t_n})\) under \(\mathbb{Q}\) is Gaussian. Its mean vector has entries \(A(t_i)\), and its covariance matrix has entries \(\min(t_i,t_j)\). Thus a deterministic drift change translates the Gaussian vector but leaves its covariance matrix unchanged. Equivalently, for each interval \([s,t]\), the \(\mathbb{Q}\)-mean of \(B_t-B_s\) is \(\int_s^t\theta(u)\,du\), while its variance remains \(t-s\).

Common mistake: reversing the density sign. With \(Z_T=\exp(\int_0^T\theta\,dB-\tfrac12\int_0^T\theta^2dt)\), the transformed Brownian motion is \(B_t-\int_0^t\theta(s)\,ds\). For example, if \(\theta(t)=t\) on \([0,2]\), the mean of \(B_2\) under this measure is \(A(2)=\int_0^2t\,dt=2\), not \(-2\). A density with \(-\int\theta\,dB\) instead produces the opposite shift. The sign must be read from the chosen density, not recalled independently.

The finite-horizon square-integrability assumption is important: it makes the deterministic stochastic integral well-defined and gives the finite variance \(v(T)\). The result concerns a change from \(\mathbb{P}\) to the explicitly defined \(\mathbb{Q}\); it does not identify \(\mathbb{P}\) with a risk-neutral measure or imply that every adapted, random drift can be handled by the same direct Gaussian calculation. In particular, for random adapted drifts the density’s martingale property requires separate justification, such as an applicable condition in Girsanov’s theorem.