Girsanov's Theorem: Changing the Drift of Brownian Motion

Precise Statement

Fix a filtered probability space (Ω, ℱ, (ℱₜ)_{0≤t≤T}, ℙ) satisfying the usual conditions (right-continuity, ℙ-completeness), and let Wₜ = (Wₜ¹,…,Wₜᵈ) be a d-dimensional (ℱₜ)-Brownian motion under ℙ. Let θₜ = (θₜ¹,…,θₜᵈ) be an (ℱₜ)-progressively measurable process with ∫₀ᵀ‖θₛ‖² ds < ∞ almost surely. Define the stochastic exponential

Zₜ = ℰ(−∫θ·dW)ₜ = exp( −∫₀ᵗ θₛ·dWₛ − ½∫₀ᵗ ‖θₛ‖² ds ).

If Zₜ is a true martingale on [0,T] (so 𝔼ℙ[Z_T] = 1), define an equivalent measure ℚ on ℱ_T by dℚ/dℙ = Z_T. Then Girsanov's theorem asserts that the process

W̃ₜ = Wₜ + ∫₀ᵗ θₛ ds, 0 ≤ t ≤ T,

is a standard d-dimensional Brownian motion under ℚ. The drift ∫θ ds present under ℙ is exactly compensated by the change of measure, and ℙ, ℚ are mutually equivalent on ℱ_T.

The Picture: Reweighting Paths, Not Moving Them

The subtle point is that Girsanov does not transform paths. The sample function t ↦ Wₜ(ω) is untouched; what changes is how much probability mass we assign to each ω. Think of ℙ as a cloud of trajectories, symmetric around zero drift. To simulate a positive drift, we don't push the paths upward — we simply declare that the already-existing upward-wandering paths are more likely, downweighting the downward ones, via the multiplier Z_T.

The density Zₜ = exp(−∫θ·dW − ½∫‖θ‖²) is the unique reweighting that preserves the quadratic variation ⟨W̃⟩ₜ = t·I_d while injecting exactly the drift θ. The −½∫‖θ‖² term is a normalization: it is precisely the Itô correction that keeps 𝔼ℙ[Zₜ] = 1, so ℚ remains a probability measure. Cameron and Martin (1944) discovered the special case of a deterministic constant shift; Girsanov (1960) allowed θ to be a random, adapted, path-dependent process — that generality is what makes the theorem indispensable for stochastic control and finance.

Key Idea of the Proof

The engine is Lévy's characterization: a continuous local martingale M with M₀ = 0 and ⟨Mⁱ,Mʲ⟩ₜ = δᵢⱼ t is a Brownian motion. So it suffices to show W̃ₜ is a continuous ℚ-local martingale with the right bracket.

First, Zₜ solves the SDE dZₜ = −Zₜ θₜ·dWₜ (apply Itô to the exponential; the drift term cancels by construction), so Z is a ℙ-local martingale — and a true martingale under Novikov. Second, one uses the abstract Bayes rule: an adapted process Yₜ is a ℚ-martingale iff Yₜ Zₜ is a ℙ-martingale. Apply this to Yₜ = W̃ₜ. By Itô's product rule, d(W̃Z) = Z dW̃ + W̃ dZ + d⟨W̃,Z⟩. Now d⟨W̃,Z⟩ₜ = d⟨W,Z⟩ₜ = −Zₜ θₜ dt (since ⟨∫θ ds, Z⟩ = 0), and dW̃ = dW + θ dt. The θ dt drift from dW̃ and the −θ dt cross-variation cancel exactly, leaving W̃Z a ℙ-local martingale. Hence W̃ is a ℚ-local martingale. Its bracket ⟨W̃ⁱ,W̃ʲ⟩ = ⟨Wⁱ,Wʲ⟩ = δᵢⱼ t is measure-invariant. Lévy finishes it.

Canonical Example: Constant Drift (Cameron–Martin)

Take d = 1, θₜ ≡ μ constant, on [0,T]. Then Novikov holds trivially since 𝔼[exp(½μ²T)] < ∞, so Z is a true martingale:

Z_T = exp(−μ W_T − ½ μ² T).

Under ℚ with dℚ = Z_T dℙ, the process W̃ₜ = Wₜ + μt is a standard Brownian motion. Read this backward: under ℙ, W̃ₜ is Brownian motion with drift μ. So Girsanov relates the law of drifted BM to that of driftless BM by an explicit exponential density. This underlies the exact formula for the distribution of the maximum of drifted Brownian motion and hitting-time densities. Concretely, one can compute ℙ(W̃_T ∈ A) = 𝔼ℚ[𝟙_{W̃_T∈A} / Z_T] = 𝔼ℚ[𝟙_{W̃_T∈A} exp(μ W̃_T − ½μ²T)] — the Gaussian tilt exp(μx − ½μ²T) shifts a mean-0 Gaussian to a mean-μT Gaussian, recovering the elementary fact that adding drift μt shifts the marginal mean by μT. The whole diffusion-scale structure follows from this one identity.

Why the Hypotheses Matter

The delicate hypothesis is that Z must be a true martingale, not merely a local one. Z is always a nonnegative local martingale, hence a supermartingale, so 𝔼ℙ[Z_T] ≤ 1 automatically. The failure mode is strict inequality: if 𝔼ℙ[Z_T] < 1, then dℚ = Z_T dℙ has total mass < 1 and is not a probability measure — the drift removal breaks. Novikov's condition 𝔼ℙ[exp(½∫₀ᵀ‖θₛ‖²ds)] < ∞ is the standard sufficient guarantee; Kazamaki's condition, phrased on the local-martingale part M = −∫θ·dW of Z (namely that exp(½M_T) = exp(−½∫θ·dW) be a uniformly integrable submartingale, equivalently sup_τ 𝔼[exp(½M_τ)] < ∞), is weaker. A classic counterexample: let θₜ depend on a process that explodes fast enough that 𝔼[Z_T] < 1 — e.g. drifts growing like the reciprocal of a Bessel process hitting 0 — and Girsanov genuinely fails. Also essential: equivalence only holds on ℱ_T for finite T. On the infinite horizon or the full path σ-algebra, ℙ and ℚ can become mutually singular (drifted and driftless BM have singular laws on C[0,∞)), so no Radon–Nikodym density exists.

Applications and Significance

Girsanov is the mathematical heart of arbitrage-free pricing. The Fundamental Theorem of Asset Pricing says no-arbitrage ≈ existence of an equivalent martingale measure ℚ under which discounted asset prices are martingales; Girsanov constructs exactly this ℚ by choosing θ = the market price of risk to cancel the real-world drift. Black–Scholes prices are ℚ-expectations, and the same tilt converts a stock's true growth rate into the risk-free rate.

In nonlinear filtering, the Kallianpur–Striebel formula and Zakai equation reference-measure trick use Girsanov to reduce a signal-plus-noise observation to independent noise. In large deviations, Schilder's theorem and the Freidlin–Wentzell theory tilt the driftless measure to make rare drifted paths typical, with the −½∫‖θ‖² term becoming the rate functional. It also grounds importance sampling for SDEs, Maximum-likelihood estimation of diffusion drift (the likelihood ratio is Z_T), and the well-posedness of SDEs with irregular drift via the Zvonkin–Veretennikov removal-of-drift method.

How the drift θ is absorbed: quantities under ℙ versus under the tilted measure ℚ

Object	Under ℙ (original)	Under ℚ (dℚ = Z_T dℙ)
W̃ₜ = Wₜ + ∫₀ᵗθₛds	Brownian motion with drift ∫θ ds	Standard Brownian motion
Wₜ	Standard Brownian motion	Brownian motion with drift −∫θ ds
Density Zₜ	ℙ-martingale, Z₀ = 1, 𝔼ℙ[Zₜ] = 1	dℚ/dℙ restricted to ℱₜ
Quadratic variation ⟨W̃⟩ₜ	t·I_d	t·I_d (unchanged)
Measures ℙ, ℚ	—	Mutually equivalent on ℱ_T (finite T)
Novikov 𝔼[e^{½∫‖θ‖²}] < ∞	Guarantees Z is a true (not just local) martingale	Ensures 𝔼ℙ[Z_T]=1 so ℚ is a probability measure