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Girsanov's theorem describes how Brownian motion transforms under a change of probability measure. If W is a Brownian motion under P and we define a new measure Q via the Radon-Nikodym derivative dQ/dP = exp(-∫θ dW - (1/2)∫θ² dt), then W̃(t) = W(t) + ∫₀ᵗ θ(s) ds is a Brownian motion under Q. This allows removing or adding drift: a process with drift under one measure becomes driftless under another. It is the mathematical foundation of risk-neutral pricing in finance.
Girsanov's theorem answers a remarkable question: if you change the probability measure on a probability space, what happens to the Brownian motion? The answer is that it acquires (or loses) a drift. Specifically, if W is a standard Brownian motion under probability measure P, and we define a new measure Q by the Radon-Nikodym derivative dQ/dP = Z(T), where Z(t) = exp(-∫₀ᵗ θ(s) dW(s) - (1/2)∫₀ᵗ θ(s)² ds) is the exponential martingale, then the process W̃(t) = W(t) + ∫₀ᵗ θ(s) ds is a standard Brownian motion under Q.
The exponential Z(t) is called the Girsanov density or likelihood ratio process. Your prerequisite on the Radon-Nikodym theorem ensures you understand what dQ/dP means: it is a non-negative measurable function that converts P-expectations to Q-expectations via E_Q[X] = E_P[Z·X]. The Novikov condition E_P[exp((1/2)∫₀ᵀ θ² dt)] < ∞ is the standard sufficient condition ensuring Z is a true martingale (E_P[Z(T)] = 1), so that Q is a genuine probability measure equivalent to P. Without this condition, Z could be a strict supermartingale with E[Z(T)] < 1, and Q would assign total mass less than 1 — a defective measure.
The practical power of Girsanov's theorem is drift removal. If under P we have dX = μ(t)dt + σ(t)dW, we can choose θ = μ/σ and switch to a measure Q under which dX = σ dW̃ — the drift has been absorbed into the new Brownian motion. This is the mathematical content of risk-neutral pricing in finance: under the real-world measure P, a stock has drift μ (its expected return). Under the risk-neutral measure Q (constructed via Girsanov with θ = (μ-r)/σ, where r is the risk-free rate), the stock has drift r. Option prices are expectations under Q, not P — Girsanov's theorem is the bridge between the physical and risk-neutral worlds.
A critical limitation: Girsanov's theorem changes drift but not volatility. The quadratic variation [X,X]_t is the same under both P and Q because equivalent measures agree on null sets, and quadratic variation is determined pathwise. This means the "roughness" of sample paths is an absolute property — no change of measure can smooth Brownian motion or eliminate diffusion. Drift is a statistical property (it determines which direction the process tends to go), while volatility is a pathwise property (it determines how rough the paths are). Girsanov lets you manipulate the former while the latter remains invariant.