Under measure P, the process X(t) satisfies dX = μ dt + σ dW where W is a P-Brownian motion. Using Girsanov's theorem with θ = μ/σ, under the new measure Q:
BX(t) satisfies dX = σ dW̃ — the drift is absorbed into the new Brownian motion W̃
CX(t) satisfies dX = μ dt + σ dW̃ — only the Brownian motion changes, not the drift
DX(t) becomes deterministic under Q
With θ = μ/σ, Girsanov's theorem says W̃(t) = W(t) + (μ/σ)t is a Q-Brownian motion. Rewriting: W(t) = W̃(t) - (μ/σ)t. Substituting into dX: dX = μ dt + σ(dW̃ - (μ/σ)dt) = μ dt + σ dW̃ - μ dt = σ dW̃. The drift μ dt has been completely absorbed. Under Q, X is a driftless diffusion — a scaled Brownian motion. This is the core mechanism of risk-neutral pricing: the real-world drift μ of a stock becomes the risk-free rate r after the appropriate change of measure.
Question 2 Multiple Choice
The Novikov condition E_P[exp((1/2)∫₀ᵀ θ²(t) dt)] < ∞ is a sufficient condition for:
AThe SDE dX = θX dt + dW to have a unique solution
BThe exponential local martingale Z(t) = exp(-∫₀ᵗ θ dW - (1/2)∫₀ᵗ θ² ds) to be a true martingale, ensuring the Girsanov change of measure is valid
CThe process W̃(t) to have independent increments under P
DThe drift θ(t) to be bounded almost surely
The Girsanov density Z(t) = exp(-∫θ dW - (1/2)∫θ² ds) is always a non-negative local martingale, hence a supermartingale with E[Z(t)] ≤ 1. For Girsanov's theorem to work, Z must be a true martingale (E[Z(T)] = 1), so that dQ = Z(T) dP defines a genuine probability measure. Novikov's condition provides the sufficient integrability to upgrade the local martingale to a true martingale. Without it, the 'measure' Q might have total mass less than 1 and fail to be a probability measure.
Question 3 Short Answer
Explain why Girsanov's theorem does not allow you to change a Brownian motion into a deterministic process, even though it can remove drift.
Think about your answer, then reveal below.
Model answer: Girsanov's theorem changes the drift of a process but not its diffusion coefficient. Under any equivalent measure Q, the quadratic variation of the process is unchanged ([X,X]_t is the same under P and Q), so the noise σ dW̃ persists — only its distribution relative to the new Brownian motion W̃ is reinterpreted. Removing drift converts X from 'Brownian motion plus drift' to 'Brownian motion,' not to a constant. To make the process deterministic, you would need to eliminate the diffusion coefficient σ, which no absolutely continuous change of measure can do — singular measures (with dQ/dP = 0 on some events) would be required.
This reflects a deep principle: equivalent probability measures (P ~ Q with dQ/dP > 0 a.s.) agree on which events have probability zero. A Brownian path is almost surely non-differentiable under P, and this remains true under any equivalent Q. You can change what looks like drift but not the fundamental roughness of the paths.