Chapter 10

Rigorous

Girsanov's Theorem and Change of Measure

00 · Symbol Glossary

$\mathbb{P},\ \mathbb{Q}$P and Q — two probability measures

Two ways of assigning probabilities to the same set of paths. $\mathbb{P}$ is usually the real-world (physical) measure; $\mathbb{Q}$ is a reweighted measure built for pricing. Both live on the same sample space and agree on which paths are possible — only the weight placed on each path differs.

$Z_T=\frac{d\mathbb{Q}}{d\mathbb{P}}$Radon–Nikodym derivative

A positive random variable that converts $\mathbb{P}$ -probabilities into $\mathbb{Q}$ -probabilities: $\mathbb{E}^{\mathbb{Q}}[X]=\mathbb{E}^{\mathbb{P}}[XZ_T]$ for every $X$ . Requires $Z_T>0$ and $\mathbb{E}^{\mathbb{P}}[Z_T]=1$ , so $\mathbb{Q}$ is a genuine probability measure.

$Z_t$density process

The $\mathbb{P}$ -martingale $Z_t=\mathbb{E}^{\mathbb{P}}[Z_T\mid\mathcal{F}_t]$ (Chapter 04) that carries the reweighting through time. $Z_0=1$ , and $Z_t$ tells you the running likelihood ratio using only information available up to $t$ .

$\theta_t$Girsanov kernel

The adapted process controlling how much drift the change of measure absorbs. In the finance application below, $\theta=(\mu-r)/\sigma$ — excess return per unit of volatility.

$\tilde W_t$W tilde — Brownian motion under Q

$\tilde W_t=W_t+\int_0^t\theta_s\,ds$ . Under $\mathbb{Q}$ , $\tilde W_t$ has exactly the defining properties Chapter 02 gave $W_t$ under $\mathbb{P}$ : continuous paths, independent increments, $\tilde W_t-\tilde W_s\sim N(0,t-s)$ .

01 · Same Noise, Different Drift

An SDE $dX_t=\mu_t\,dt+\sigma_t\,dW_t$ (Chapter 07) has two ingredients that feel similar but behave completely differently under a change of probability measure. Quadratic variation (Chapter 03) is computed from a single observed path — sum the squared increments along the trajectory that actually happened, and the limit is $\int_0^t\sigma_s^2\,ds$ , no probabilities involved. Drift is the opposite: $\mu_t=\lim_{h\to0}\mathbb{E}[X_{t+h}-X_t]/h$ is an average over paths, and an average is exactly the kind of object that changes when you reweight which paths count more.

That asymmetry is the entire content of this chapter. Reweighting probabilities — assigning more mass to some paths and less to others, without altering the set of paths themselves — can turn one drift into a different drift on the same noise. It cannot touch volatility, because volatility is read off a single path and a change of measure never edits a path, only how likely it was.

A crowd, not a coin

Think of $\mathbb{P}$ and $\mathbb{Q}$ as two different opinions about how likely each possible path is, held by two observers watching the same physical process unfold. Neither observer can change what actually happens. They can only disagree about which outcomes were probable in advance — which is precisely enough to disagree about means, but not enough to disagree about a quantity like quadratic variation that is determined by the realized path alone.

Practice — Which quantities can a measure change touch?

For $dX_t=\mu_t\,dt+\sigma_t\,dW_t$ , list which of the following could differ between an equivalent $\mathbb{P}$ and $\mathbb{Q}$ : the realized path itself; $\mathbb{E}^{\mathbb{P}}[X_t]$ versus $\mathbb{E}^{\mathbb{Q}}[X_t]$ ; the quadratic variation $[X]_t$ ; the set of paths assigned probability zero. Justify each answer in one sentence.

02 · The Radon–Nikodym Derivative, Informally

Definition — Equivalent Measures

Two measures $\mathbb{P}$ and $\mathbb{Q}$ on the same space are equivalent ( $\mathbb{P}\sim\mathbb{Q}$ ) if they agree on which events are impossible: $\mathbb{P}(A)=0 \iff \mathbb{Q}(A)=0$ . Equivalence (not just similarity) is exactly the condition under which a Radon–Nikodym derivative $Z_T=d\mathbb{Q}/d\mathbb{P}$ exists: a nonnegative random variable with $\mathbb{E}^{\mathbb{P}}[Z_T]=1$ such that

\mathbb{E}^{\mathbb{Q}}[X] = \mathbb{E}^{\mathbb{P}}[X\,Z_T] \quad \text{for every random variable } X

$Z_T$ is a likelihood ratio: on a path where $Z_T$ is large, $\mathbb{Q}$ thinks that path was more likely than $\mathbb{P}$ did, and vice versa. Because expectations under $\mathbb{Q}$ are just $\mathbb{P}$ -expectations with an extra factor of $Z_T$ inside, every $\mathbb{Q}$ -probability statement can be computed by staying entirely inside $\mathbb{P}$ and multiplying by $Z_T$ .

Example — A two-path cartoon

Suppose only two paths are possible, $\omega_1$ and $\omega_2$ , with $\mathbb{P}(\omega_1)=0.5$ , $\mathbb{P}(\omega_2)=0.5$ . Define $\mathbb{Q}(\omega_1)=0.8$ , $\mathbb{Q}(\omega_2)=0.2$ . Then $Z(\omega_1)=0.8/0.5=1.6$ and $Z(\omega_2)=0.2/0.5=0.4$ , and $\mathbb{E}^{\mathbb{P}}[Z]=0.5(1.6)+0.5(0.4)=1$ as required. Nothing about $\omega_1$ or $\omega_2$ as paths changed — only how much probability mass sits on each one.

Why Z_t must be a martingale

Consistency across time forces $Z_t=\mathbb{E}^{\mathbb{P}}[Z_T\mid\mathcal{F}_t]$ to be a $\mathbb{P}$ -martingale: today's best guess of the eventual likelihood ratio, updated as information arrives, is by construction unbiased under $\mathbb{P}$ (the tower property of conditional expectation, Chapter 04). This is what lets the measure change be applied gradually along a filtration rather than only at the terminal time $T$ .

03 · Girsanov's Theorem: Statement and Payoff

Definition — Girsanov's Theorem

Let $\theta_t$ be adapted and satisfy Novikov's condition ( $\mathbb{E}^{\mathbb{P}}[\exp(\tfrac12\int_0^T\theta_s^2\,ds)]<\infty$ , enough to rule out pathological blow-ups). Define the density process

Z_t = \exp\left(-\int_0^t\theta_s\,dW_s - \frac12\int_0^t\theta_s^2\,ds\right)

Then $Z_t$ is a $\mathbb{P}$ -martingale, and setting $d\mathbb{Q}=Z_T\,d\mathbb{P}$ defines an equivalent measure $\mathbb{Q}$ under which

\tilde W_t = W_t + \int_0^t\theta_s\,ds

is a standard Brownian motion.

Rearranged, $dW_t = d\tilde W_t - \theta_t\,dt$ . Substitute this into any SDE written in terms of $W_t$ and the diffusion term is untouched — $\sigma_t\,dW_t=\sigma_t\,d\tilde W_t-\sigma_t\theta_t\,dt$ moves a piece from noise into drift, but the coefficient multiplying the new Brownian motion is still exactly $\sigma_t$ . This is the payoff: Girsanov lets you dial the drift of a diffusion to any adapted process you like by an appropriate choice of $\theta_t$ , while the volatility $\sigma_t$ is invariant — it appears unchanged on both sides of the substitution.

❌ Expecting Girsanov to change volatility

Hoping that some clever choice of $\theta_t$ could turn $dX_t=\mu\,dt+\sigma\,dW_t$ into a process with a different diffusion coefficient, say $\sigma'\neq\sigma$ , purely by a change of measure.

Why it breaks: equivalent measures share null sets by definition, and quadratic variation is a pathwise a.s. limit (Chapter 03). If $\mathbb{P}$ and $\mathbb{Q}$ disagreed about $[X]_t=\sigma^2 t$ , that disagreement would itself be an event of probability $0$ under one measure and positive probability under the other — contradicting equivalence.

Consequence: $\theta_t$ can only ever move terms between the $dt$ (drift) and $dW_t$ (noise) pieces of the same $\sigma_t\,dW_t$ ; it can never rescale $\sigma_t$ itself. A model that needs different volatility is a different model, not a different measure.

04 · Worked Example: Removing the Drift from GBM

Step-by-step — Shifting GBM's drift to r via Girsanov

Start from the GBM SDE (Chapter 08):

dS_t=\mu S_t\,dt+\sigma S_t\,dW_t

, with constants

\mu,\sigma

and a constant risk-free rate

r

Choose the constant kernel

\theta=\dfrac{\mu-r}{\sigma}

, and define

\tilde W_t=W_t+\theta t

, a

\mathbb{Q}

-Brownian motion by Girsanov.

Substitute

dW_t=d\tilde W_t-\theta\,dt

into the SDE:

dS_t=\mu S_t\,dt+\sigma S_t(d\tilde W_t-\theta\,dt)=(\mu-\sigma\theta)S_t\,dt+\sigma S_t\,d\tilde W_t

Simplify the drift:

\mu-\sigma\theta=\mu-\sigma\cdot\dfrac{\mu-r}{\sigma}=\mu-(\mu-r)=r

Conclude: under

\mathbb{Q}

dS_t=rS_t\,dt+\sigma S_t\,d\tilde W_t

— same GBM, same

\sigma

, drift replaced by

r

Nothing about the asset's actual path changed; $\mathbb{Q}$ simply weights paths differently so that, averaged under this new weighting, $S_t$ grows at the risk-free rate. A short calculation with Itô's lemma (Chapter 06) on $e^{-rt}S_t$ confirms this drift shift makes the discounted price a $\mathbb{Q}$ -martingale — the property that later chapters build derivative pricing on.

This measure has a name

$\mathbb{Q}$ here is the risk-neutral measure, and $\theta=(\mu-r)/\sigma$ is the market price of risk. Nothing about risk preferences enters the mathematics of this section — the name describes how the resulting measure gets used (Chapters 12–13), not a step in the derivation.

05 · Exercises

EXERCISE 10.1

Apply Itô's lemma (Chapter 06) to $\ln Z_t$ , where $Z_t=\exp(-\int_0^t\theta_s\,dW_s-\tfrac12\int_0^t\theta_s^2\,ds)$ .

Let $Y_t=-\int_0^t\theta_s\,dW_s-\tfrac12\int_0^t\theta_s^2\,ds$ , so $Z_t=e^{Y_t}$ with $dY_t=-\theta_t\,dW_t-\tfrac12\theta_t^2\,dt$ . Itô's lemma gives $dZ_t=Z_t\,dY_t+\tfrac12Z_t\,(dY_t)^2=Z_t(-\theta_t\,dW_t-\tfrac12\theta_t^2\,dt)+\tfrac12Z_t\theta_t^2\,dt=-Z_t\theta_t\,dW_t$ . The $dt$ terms exactly cancel, leaving pure diffusion with no drift — $Z_t$ is (at least) a local martingale, and Novikov's condition upgrades this to a true martingale.

Show that the density process $Z_t$ has zero drift, i.e. that $dZ_t=-Z_t\theta_t\,dW_t$ with no $dt$ term.

EXERCISE 10.2

Quadratic variation is computed from squared increments of the realized path; check that substituting $dW_t=d\tilde W_t-\theta\,dt$ does not change the $\sigma^2S_t^2\,dt$ piece.

$[S]_t=\int_0^t\sigma^2S_s^2\,ds$ under either the $\mathbb{P}$ -representation $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$ or the $\mathbb{Q}$ -representation $dS_t=rS_t\,dt+\sigma S_t\,d\tilde W_t$ , because quadratic variation only sees the coefficient multiplying the Brownian term, and that coefficient is $\sigma S_t$ in both. The $dt$ terms (drift) never contribute to quadratic variation regardless of which measure produced them.

Confirm that $S_t$ 's quadratic variation is the same whether computed from the $\mathbb{P}$ -SDE or the $\mathbb{Q}$ -SDE of Section 04.

EXERCISE 10.3

Compute the quadratic variation of $cW_t$ for a constant $c\neq1$ and compare it to that of $W_t$ .

$[cW]_t=c^2t$ , while $[W]_t=t$ . If some measure change turned $W_t$ (under $\mathbb{P}$ ) into $cW_t$ (under an alleged $\mathbb{Q}$ ) with $c\neq1$ , then $\mathbb{P}$ and $\mathbb{Q}$ would disagree about the value of a pathwise-computable quantity ( $t$ versus $c^2t$ ) on a set of full probability — precisely the disagreement equivalence rules out. So no change of measure can rescale Brownian motion; Girsanov only ever shifts drift.

Explain, using quadratic variation, why no Girsanov-type change of measure can map $W_t$ to $cW_t$ for a constant $c\neq1$ .

06 · Chapter Summary

Concept	Meaning
Equivalent measures	$\mathbb{P}\sim\mathbb{Q}$ : same null sets; $Z_T=d\mathbb{Q}/d\mathbb{P}$ exists
Density process $Z_t$	$\mathbb{P}$ -martingale carrying the reweighting through time
Girsanov	$dW_t=d\tilde W_t-\theta_t\,dt$ ; reshapes drift only
$\theta=(\mu-r)/\sigma$	shifts GBM's drift from $\mu$ to $r$ under $\mathbb{Q}$
Invariant	$\sigma_t$ and quadratic variation — identical under $\mathbb{P}$ and $\mathbb{Q}$
Changed	drift $\mu_t\to\mu_t-\sigma_t\theta_t$

Next: Chapter 11 — Feynman–Kac and the Black-Scholes PDE, where the risk-neutral drift built here turns a pricing expectation into a partial differential equation.