Change of Measure and Girsanov's Theorem

Replacing one measure by an equivalent one reweights probability without altering which events are negligible. On a filtered space the reweighting is dynamic, carried by a positive martingale whose terminal value is the Radon-Nikodym density. The same device removes a drift from a Brownian motion and converts a model under the physical measure into a model under which discounted prices are martingales.

#The density process

Let $\Q\sim\P$ on $\F_T$ with density $Z_T=\dd\Q/\dd\P$ , a positive integrable random variable with $\E_\P[Z_T]=1$ . Define the density process $Z_t=\E_\P[Z_T\mid\F_t]$ for $t\le T$ .

Proposition1

The density process $Z$ is a positive $\P$ -martingale, and on $\F_t$ the restriction of $\Q$ has $\P$ -density $Z_t$ , that is $\Q(A)=\E_\P[Z_t\,\mathbf 1_A]$ for $A\in\F_t$ .

Proof

As a conditional expectation of a fixed integrable variable, $Z$ is a martingale by the tower property, $\E_\P[Z_t\mid\F_s]=\E_\P[\E_\P[Z_T\mid\F_t]\mid\F_s]=\E_\P[Z_T\mid\F_s]=Z_s$ . It is strictly positive, for if $A=\{Z_t=0\}\in\F_t$ then $0=\E_\P[Z_t\,\mathbf 1_A]=\E_\P[Z_T\,\mathbf 1_A]$ , which with $Z_T>0$ a.s. forces $\P(A)=0$ , so $Z_t>0$ $\P$ -a.s. and the division by $Z_s$ below is legitimate. For $A\in\F_t$ the defining property of conditional expectation gives $\E_\P[Z_t\,\mathbf 1_A]=\E_\P[Z_T\,\mathbf 1_A]=\E_\Q[\mathbf 1_A]=\Q(A)$ , where the middle equality is the defining change of measure on $\F_T$ , valid for $A$ because $A\in\F_t\subseteq\F_T$ .

#The Bayes rule

Theorem2

For $0\le s\le t\le T$ and an $\F_t$ -measurable integrable $X$ ,

\E_\Q[X\mid\F_s]=\frac{\E_\P[Z_t\,X\mid\F_s]}{Z_s}\qquad\Q\text{-almost surely.} \tag{1}

Proof

The right side is $\F_s$ -measurable, so it suffices to check the averaging identity over an arbitrary $A\in\F_s$ . Extending Proposition 1 from indicators to $\F_t$ -measurable $Y\ge0$ by monotone convergence, and then to integrable $Y$ by splitting into positive and negative parts, gives $\E_\Q[Y]=\E_\P[Z_t Y]$ ; here $Y=\mathbf 1_A X$ with $\E_\P[Z_t|X|]=\E_\Q[|X|]<\infty$ , so every term is finite. Using this at level $t$ , then the tower property,

\E_\Q[\mathbf 1_A X]=\E_\P[Z_t\,\mathbf 1_A X]=\E_\P\big[\mathbf 1_A\,\E_\P[Z_t X\mid\F_s]\big]=\E_\P\!\left[\mathbf 1_A\,Z_s\,\frac{\E_\P[Z_t X\mid\F_s]}{Z_s}\right], \tag{2}

and the last expression equals $\E_\Q\big[\mathbf 1_A\,\E_\P[Z_t X\mid\F_s]/Z_s\big]$ by Proposition 1 at level $s$ . The two sides agree over every $A\in\F_s$ , which is Equation (1).

#Girsanov's theorem

The density that removes a drift is a Doleans (stochastic) exponential. For a predictable $\theta$ with $\int_0^T\theta_s^2\,ds<\infty$ a.s., so that the Ito integral and the exponential are defined, the Doleans exponential of the Brownian integral is

Z_t=\exp\!\left(\int_0^t\theta_s\,dW_s-\tfrac12\int_0^t\theta_s^2\,ds\right), \tag{3}

the unique solution of $dZ_t=Z_t\,\theta_t\,dW_t$ . It is a positive local martingale, and a true martingale under the Novikov condition $\E_\P[\exp(\tfrac12\int_0^T\theta_s^2\,ds)]<\infty$ [1].

Theorem3

Let $W$ be a $\P$ -Brownian motion and let $Z$ from Equation (3) be a martingale on $[0,T]$ . Define $\Q$ by $\dd\Q/\dd\P=Z_T$ . Then

\widetilde W_t=W_t-\int_0^t\theta_s\,ds \tag{4}

is a Brownian motion under $\Q$ .

Proof

By Levy's characterization it suffices that $\widetilde W$ is a continuous $\Q$ -local martingale with $\langle\widetilde W\rangle_t=t$ . Subtracting the finite-variation drift $\int_0^t\theta_s\,ds$ leaves the bracket unchanged, so $\langle\widetilde W\rangle_t=\langle W\rangle_t=t$ under $\P$ ; since quadratic variation is an a.s. pathwise limit it is invariant under the equivalent change of measure, so $\langle\widetilde W\rangle_t=t$ holds $\Q$ -a.s. as Levy under $\Q$ requires. For the martingale property, the product rule applied to $Z_t\widetilde W_t$ gives $d(Z\widetilde W)=Z\,d\widetilde W+\widetilde W\,dZ+d\langle Z,\widetilde W\rangle$ , and substituting $dZ=Z\theta\,dW$ , $d\widetilde W=dW-\theta\,dt$ , and $d\langle Z,\widetilde W\rangle=Z\theta\,dt$ shows the drift terms cancel, leaving $d(Z\widetilde W)=Z(1+\theta\widetilde W)\,dW$ , a driftless Ito integral, so $Z\widetilde W$ is a $\P$ -local martingale. Since $Z$ is a strictly positive $\P$ -martingale and $Z\widetilde W$ is a $\P$ -local martingale, $\widetilde W$ is a $\Q$ -local martingale, because multiplication by $Z$ is a bijection between $\Q$ -local martingales and $\P$ -local martingales, the local form of the Bayes correspondence of Theorem 2 [2]. That is exactly the $\Q$ -local martingale property that Levy's characterization consumes.

Choosing $\theta$ to cancel the drift of a discounted price turns that price into a $\Q$ -martingale, so $\Q$ is an equivalent martingale measure. The density $Z_T$ is the likelihood ratio between the physical and pricing measures.

[1]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed. Springer, 1991.

[2]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed. Springer, 1999.

Explore connections

see in the atlas

referenced by (1)

The Black-Scholes Equation

cite

@misc{change-of-measure,
  author = {Zac Kienzle},
  title  = {Change of Measure and Girsanov's Theorem},
  year   = {2026},
  month  = {05},
  url    = {https://zackienzle.com/blog/change-of-measure}
}

#The density process

Let $\Q\sim\P$ on $\F_T$ with density $Z_T=\dd\Q/\dd\P$ , a positive integrable random variable with $\E_\P[Z_T]=1$ . Define the density process $Z_t=\E_\P[Z_T\mid\F_t]$ for $t\le T$ .

Proposition1

The density process $Z$ is a positive $\P$ -martingale, and on $\F_t$ the restriction of $\Q$ has $\P$ -density $Z_t$ , that is $\Q(A)=\E_\P[Z_t\,\mathbf 1_A]$ for $A\in\F_t$ .

Proof

#The Bayes rule

Theorem2

For $0\le s\le t\le T$ and an $\F_t$ -measurable integrable $X$ ,

\E_\Q[X\mid\F_s]=\frac{\E_\P[Z_t\,X\mid\F_s]}{Z_s}\qquad\Q\text{-almost surely.} \tag{1}

Proof

\E_\Q[\mathbf 1_A X]=\E_\P[Z_t\,\mathbf 1_A X]=\E_\P\big[\mathbf 1_A\,\E_\P[Z_t X\mid\F_s]\big]=\E_\P\!\left[\mathbf 1_A\,Z_s\,\frac{\E_\P[Z_t X\mid\F_s]}{Z_s}\right], \tag{2}

and the last expression equals $\E_\Q\big[\mathbf 1_A\,\E_\P[Z_t X\mid\F_s]/Z_s\big]$ by Proposition 1 at level $s$ . The two sides agree over every $A\in\F_s$ , which is Equation (1).

#Girsanov's theorem

Z_t=\exp\!\left(\int_0^t\theta_s\,dW_s-\tfrac12\int_0^t\theta_s^2\,ds\right), \tag{3}

the unique solution of $dZ_t=Z_t\,\theta_t\,dW_t$ . It is a positive local martingale, and a true martingale under the Novikov condition $\E_\P[\exp(\tfrac12\int_0^T\theta_s^2\,ds)]<\infty$ [1].

Theorem3

Let $W$ be a $\P$ -Brownian motion and let $Z$ from Equation (3) be a martingale on $[0,T]$ . Define $\Q$ by $\dd\Q/\dd\P=Z_T$ . Then

\widetilde W_t=W_t-\int_0^t\theta_s\,ds \tag{4}

is a Brownian motion under $\Q$ .

Proof

[1]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed. Springer, 1991.

[2]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed. Springer, 1999.

Explore connections

see in the atlas

referenced by (1)

The Black-Scholes Equation

cite

@misc{change-of-measure,
  author = {Zac Kienzle},
  title  = {Change of Measure and Girsanov's Theorem},
  year   = {2026},
  month  = {05},
  url    = {https://zackienzle.com/blog/change-of-measure}
}