Girsanov's Theorem

Motivation: why this matters in quant finance

Girsanov's theorem is the theorem that lets a quant replace real-world drift with pricing drift by changing probability measure. In Black-Scholes language, it explains why the stock may have drift $\mu$ under the real-world measure, but drift $r$ under the risk-neutral measure, while volatility stays the same.

Lawler builds the theorem from a simple idea: there are two ways to turn a fair game into an unfair one. You can add a deterministic drift to the path, or you can leave the paths alone and change their probabilities. Girsanov says exactly how the second method works for Brownian motion.

The theorem is not just "drift changes under $\mathbb{Q}$ ." It is a precise Radon-Nikodym construction: weight Brownian paths by a nonnegative martingale, and under the weighted measure the Brownian motion acquires a drift. The martingale condition is the key technical gatekeeper.

The informal idea

Start with a Brownian motion $B_t$ under a probability measure $\mathbb{P}$ . If we define

W_t=B_t-\int_0^t A_s\,ds,

then under the original measure $W_t$ is generally not Brownian; it has had a pathwise drift removed.

Girsanov's theorem says that if we also change the probability measure using the right exponential martingale, then $W_t$ becomes Brownian under the new measure. Equivalently, under the new measure,

dB_t=A_t\,dt+dW_t.

So the same sample paths are being reweighted. The volatility, encoded in quadratic variation, does not change; the drift does.

Radon-Nikodym weights

Let $M_t$ be a nonnegative martingale with $M_0=1$ . For an event $V\in\mathcal{F}_t$ , define a new probability measure $\mathbb{P}^\ast$ on $\mathcal{F}_t$ by

\mathbb{P}^\ast(V)=\mathbb{E}^{\mathbb{P}}[1_V M_t].

Equivalently,

\frac{d\mathbb{P}^\ast}{d\mathbb{P}}\bigg|_{\mathcal{F}_t}=M_t.

The martingale property makes this definition consistent through time. If $V\in\mathcal{F}_s$ and $s<t$ , then

\mathbb{E}[1_VM_t] =\mathbb{E}[1_V\mathbb{E}(M_t\mid\mathcal{F}_s)] =\mathbb{E}[1_VM_s].

This is why Girsanov is a martingale theorem as much as a change-of-measure theorem.

The exponential martingale

Lawler states Girsanov for a nonnegative martingale satisfying the exponential SDE

dM_t=A_tM_t\,dB_t, \qquad M_0=1.

Formally, the solution is

M_t=\exp\left\{\int_0^t A_s\,dB_s-\frac{1}{2}\int_0^t A_s^2\,ds\right\}.

This formula comes from Itô's lemma: the

-\frac{1}{2}\int A_s^2\,ds

term is exactly the correction needed to remove drift from the exponential.

Formal statement

Suppose $B_t$ is Brownian motion under $\mathbb{P}$ , and suppose $M_t$ is a nonnegative martingale satisfying

dM_t=A_tM_t\,dB_t, \qquad M_0=1.

Define $\mathbb{P}^\ast$ by

\frac{d\mathbb{P}^\ast}{d\mathbb{P}}\bigg|_{\mathcal{F}_t}=M_t.

Then

W_t=B_t-\int_0^t A_s\,ds

is a standard Brownian motion under $\mathbb{P}^\ast$ . In differential notation,

dB_t=A_t\,dt+dW_t.

The theorem says that weighting by $M_t$ gives $B_t$ drift $A_t$ in the new measure.

Constant drift as the basic example

For constant $m$ , take

M_t=\exp\left(mB_t-\frac{1}{2}m^2t\right).

This is a martingale satisfying

dM_t=mM_t\,dB_t.

Under the tilted measure $\mathbb{Q}$ with density $M_t$ , the process

W_t=B_t-mt

is Brownian. So $B_t$ itself is Brownian motion with drift $m$ under $\mathbb{Q}$ .

This is the continuous-time analogue of changing a binomial model's up/down probabilities so that the mean increment shifts while the step size stays the same.

Localisation and Novikov's condition

The exponential formula always gives a nonnegative local martingale, but not necessarily a true martingale. If it is not a martingale, it cannot define a probability measure with total mass one on the horizon.

Lawler handles this by localisation. Stop before either the exponential weight or the accumulated variance becomes too large, apply the theorem to the stopped square-integrable martingale, and then pass to a limiting stopping time.

A practical sufficient condition for the exponential local martingale to be a true martingale is Novikov's condition:

\mathbb{E}\left[\exp\left(\frac{1}{2}\int_0^t A_s^2\,ds\right)\right]<\infty.

When this holds on the horizon, the Girsanov density is a genuine martingale and the measure change is valid there.

Black-Scholes drift change

Suppose a stock follows

dS_t=S_t[\mu\,dt+\sigma\,dB_t]

under $\mathbb{P}$ , and a bond grows at rate $r$ . To make the discounted stock a martingale, we want a new Brownian motion $W_t$ such that

dS_t=S_t[r\,dt+\sigma\,dW_t]

under the pricing measure.

Set

A=\frac{\mu-r}{\sigma}.

dB_t=A\,dt+dW_t,

then

dS_t=S_t[\mu\,dt+\sigma(dW_t-A\,dt)] =S_t[r\,dt+\sigma\,dW_t].

This is the drift replacement Lawler uses in the martingale approach to Black-Scholes. The pricing measure is equivalent to the original measure when the exponential martingale defines the Radon-Nikodym derivative, so probability-zero events remain probability-zero events.

Worked examples

Example 1: Brownian drift from path weighting

Let $m=0.3$ and $T=1$ . The density

M_1=\exp(0.3B_1-0.045)

overweights paths with larger $B_1$ and underweights paths with smaller $B_1$ . Under the new measure, $B_t$ has drift $0.3$ and $B_1\sim N(0.3,1)$ .

Example 2: removing stock drift

If $\mu=0.10$ , $r=0.04$ , and $\sigma=0.20$ , then

A=\frac{0.10-0.04}{0.20}=0.30.

Under the tilted measure,

W_t=B_t-0.30t

is Brownian, and the stock SDE becomes

dS_t=S_t[0.04\,dt+0.20\,dW_t].

This is the risk-neutral stock dynamics used for pricing.

Example 3: why the martingale check matters

For unbounded $A_t$ , the exponential process

\exp\left\{\int_0^t A_s\,dB_s-\frac{1}{2}\int_0^t A_s^2\,ds\right\}

may be only a local martingale. Lawler's examples with Bessel-type processes show that such weights can explode or lose mass in finite time under the tilted measure. The measure change must therefore be justified, not assumed.

Common confusions and pitfalls

"Girsanov changes volatility." No. Quadratic variation is unchanged. Girsanov changes drift by reweighting paths.

"Every exponential local martingale defines a new probability measure." No. It must be a true martingale with expectation one on the horizon. Novikov is a sufficient condition.

"The Radon-Nikodym derivative is just notation." It is the object that defines the new probabilities: $\mathbb{P}^\ast(V)=\mathbb{E}[1_VM_t]$ .

"Risk-neutral drift replacement is a modelling trick." In Brownian models it is backed by Girsanov: the drift changes because the measure changes.

"Equivalent measures make all probabilities equal." They preserve which events have probability zero, not the probabilities of ordinary events.

Where this goes next

Radon-Nikodym Theorem: Supplies the density language behind measure changes.
Itô's Lemma: Produces the exponential martingale used as the Girsanov density.
Local Martingales and Semimartingales: Explains why local martingales are not automatically true martingales.
Stochastic Differential Equations: Provides the SDE form whose drift is changed.
The Derivation of the Black-Scholes Formula: Applies the drift replacement to option pricing.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 5 §5.1 (Absolutely continuous measures), §5.2 (Giving drift to a Brownian motion), §5.3 (Girsanov theorem), §5.5 (Martingale approach to Black-Scholes equation).