Geometric Brownian Motion Under a Drift Change

Changing the Drift of a Geometric Brownian Motion

Geometric Brownian motion provides a direct setting in which to see how a change of probability measure alters an SDE. The diffusion coefficient remains the same, while the drift changes by an amount determined by the volatility and the Girsanov kernel. This distinction is important in applications: the process has different distributions under the two measures, even though the measures are equivalent over a fixed finite time horizon.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\), and fix a finite horizon \(T>0\). Let \(S_0>0\), let \(\mu\in\mathbb{R}\), and let \(\sigma\ne0\). Under \(\mathbb{P}\), suppose \(S\) satisfies \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\). Here \(\mu\) is the drift coefficient and \(\sigma\) is the diffusion coefficient. Stochastic integrals in this tutorial are Ito integrals; an integral \(\int_0^t H_s\,dB_s\) is used as a square-integrable Ito integral when \(H\) is progressively measurable and \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). Under another measure \(\mathbb{Q}\), the corresponding condition is \(\mathbb{E}^{\mathbb{Q}}\int_0^T H_s^2\,ds<\infty\).

The density is well defined and the measures are equivalent on \(\mathcal{F}_T\). Indeed, \(Z_T>0\), and the constant kernel satisfies Novikov’s condition: \(\mathbb{E}\exp(\tfrac12\int_0^T\theta^2\,ds)=\exp(\tfrac12\theta^2T)<\infty\). The Girsanov theorem then applies. Its density-process and Brownian-motion conclusions are standard results of the earlier material on Girsanov’s theorem; the calculation below also directly verifies that \(Z\) is a martingale.

The Density and the Drift Transformation

For \(0\le s\le t\le T\), write \(B_t=B_s+(B_t-B_s)\). The increment \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\). For \(Y\sim N(0,v)\), its Gaussian moment-generating function gives \(\mathbb{E}[e^{cY}]=e^{c^2v/2}\). Therefore

Also \(\mathbb{E}[Z_t]=1\), by taking \(s=0\). Thus \(Z\) is a true martingale, not merely a local martingale, and its conditional expectation property is verified. The density is strictly positive, so \(\mathbb{Q}\) and \(\mathbb{P}\) are equivalent on \(\mathcal{F}_T\).

Proof. Substitute \(dB_t=dB_t^{\mathbb{Q}}-\theta\,dt\) into the original SDE. The resulting drift is \(\mu S_t-\sigma\theta S_t=\alpha S_t\), and the diffusion term is \(\sigma S_t\,dB_t^{\mathbb{Q}}\). To verify the proposed solution, set \(f(t,x)=S_0\exp((\alpha-\tfrac12\sigma^2)t+\sigma x)\). Its derivatives are \(f_t=(\alpha-\tfrac12\sigma^2)f\), \(f_x=\sigma f\), and \(f_{xx}=\sigma^2f\). Ito’s formula, including its second-derivative correction, gives \(df(t,B_t^{\mathbb{Q}})=f_t\,dt+f_x\,dB_t^{\mathbb{Q}}+\tfrac12 f_{xx}\,dt=\alpha f\,dt+\sigma f\,dB_t^{\mathbb{Q}}\). Also \(f(0,0)=S_0\). This verifies the SDE. The stochastic integral is square-integrable on \([0,T]\): the explicit formula gives \(\mathbb{E}^{\mathbb{Q}}[S_t^2]=S_0^2e^{(2\alpha+\sigma^2)t}\), so \(\mathbb{E}^{\mathbb{Q}}\int_0^T\sigma^2S_t^2\,dt<\infty\).

For uniqueness, let \(X\) be any other solution driven by the same \(B^{\mathbb{Q}}\), with the same initial value, and define \(D_t=X_t-S_t\). Then \(dD_t=\alpha D_t\,dt+\sigma D_t\,dB_t^{\mathbb{Q}}\), with \(D_0=0\). Define \(Y_t=\exp((-\alpha+\tfrac12\sigma^2)t-\sigma B_t^{\mathbb{Q}})\). Ito’s formula gives \(dY_t=Y_t((-\alpha+\sigma^2)\,dt-\sigma\,dB_t^{\mathbb{Q}})\). The Ito product rule, with \(d[Y,D]_t=-\sigma^2Y_tD_t\,dt\), now yields \(d(Y_tD_t)=0\): the diffusion terms cancel and the drift coefficient is \(\alpha+(-\alpha+\sigma^2)-\sigma^2=0\). Consequently \(Y_tD_t=Y_0D_0=0\) for all \(t\), and \(Y_t>0\) implies \(D_t=0\). This is a local calculation and uses no unproved fourth-moment bound or expectation of a local martingale. It proves pathwise uniqueness. Finally, \(\log S_t=\log S_0+(\alpha-\tfrac12\sigma^2)t+\sigma B_t^{\mathbb{Q}}\), which has the stated normal distribution because \(B_t^{\mathbb{Q}}\sim N(0,t)\) under \(\mathbb{Q}\). \(\square\)

The explicit solution is a strong solution: it is constructed from the given Brownian motion and initial value. The uniqueness argument establishes pathwise uniqueness. Its displayed normal distribution also determines its law under \(\mathbb{Q}\). These are distinct assertions: strong existence concerns construction on a given space, pathwise uniqueness compares solutions using the same Brownian motion, and uniqueness in law concerns distributions of solutions.

Worked Drift Changes and Distribution Calculations

What the Transformation Does—and Does Not Do

The diffusion coefficient \(\sigma\) is unchanged, but the drift changes from \(\mu\) to \(\mu-\sigma\theta\). With this sign convention, a positive \(\theta\) lowers the drift when \(\sigma>0\). For a desired drift \(\alpha\), the necessary constant kernel is \(\theta=(\mu-\alpha)/\sigma\), which requires \(\sigma\ne0\). If \(\sigma=0\), the process is deterministic, \(S_t=S_0e^{\mu t}\), and this Brownian density cannot change its drift.

Under \(\mathbb{Q}\), \(S\) is a martingale precisely when \(\alpha=0\). To verify this rather than infer it from a local calculation, the explicit solution and independence of Brownian increments give, for \(s\le t\), \(\mathbb{E}^{\mathbb{Q}}[S_t\mid\mathcal{F}_s]=S_s e^{\alpha(t-s)}\). The conditional expectation exists because the lognormal solution has finite first moment. It equals \(S_s\) for all \(s,t\) exactly when \(\alpha=0\). For other drift values the process is not a martingale, although its explicit formula still gives finite moments on every finite horizon.

Common mistake: reversing the sign. A frequent incorrect substitution is \(B_t=B_t^{\mathbb{Q}}+\theta t\), which would produce drift \(\mu+\sigma\theta\). But the convention \(B_t^{\mathbb{Q}}=B_t+\theta t\) implies \(B_t=B_t^{\mathbb{Q}}-\theta t\). Substituting this correct identity into \(\mu t+\sigma B_t\) gives \((\mu-\sigma\theta)t+\sigma B_t^{\mathbb{Q}}\), so the transformed drift is \(\mu-\sigma\theta\), not \(\mu+\sigma\theta\).

Another distinction is between a change of measure and a change in the process’s diffusion scale. The density tilts the distribution of Brownian paths; it does not replace \(\sigma\). Nor should an exponential local martingale automatically be treated as a probability density: a density must have expectation one. Here the constant-kernel Gaussian calculation verifies the martingale property directly, and Novikov’s condition supplies a sufficient integrability check.