In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure, which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values, to the risk-neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying.
Results of this type were first proved by Cameron-Martin in the 1940s and by Igor Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977).
Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure that is absolutely continuous with respect to P then every P-semimartingale is a Q-semimartingale.
We state the theorem first for the special case when the underlying stochastic process is a Wiener process. This special case is sufficient for risk-neutral pricing in the Black–Scholes model.
Let
\{Wt\}
\{\Omega,l{F},P\}
Xt
W | |
\{l{F} | |
t\} |
Given an adapted process
Xt
Zt=l{E}(X)t,
where
l{E}(X)
l{E}(X)t=\exp\left(Xt-
1 | |
2 |
[X]t\right),
and
[X]t
If
Zt
\{\Omega,l{F}\}
\left.
dQ | |
dP |
\right|l{Ft}=Zt=l{E}(X)t
Then for each t the measure Q restricted to the unaugmented sigma fields
o | |
l{F} | |
t |
o | |
l{F} | |
t. |
Furthermore, if
Yt
\tildeYt=Yt-\left[Y,X\right]t
is a Q local martingale on the filtered probability space
W | |
\{\Omega,F,Q,\{l{F} | |
t\}\} |
If X is a continuous process and W is Brownian motion under measure P then
\tildeWt=Wt-\left[W,X\right]t
The fact that
\tildeWt
\left[\tildeW\right]t=\left[W\right]t=t
it follows by Levy's characterization of Brownian motion that this is a Q Brownianmotion.
In many common applications, the process X is defined by
Xt=
t | |
\int | |
0 |
YsdWs.
For X of this form then a necessary and sufficient condition for
l{E}(X)
EP\left[\exp\left(
1 | |
2 |
T | |
\int | |
0 |
2 | |
Y | |
s |
ds\right)\right]<infty.
The stochastic exponential
l{E}(X)
Zt=1+
t | |
\int | |
0 |
ZsdXs.
The measure Q constructed above is not equivalent to P on
l{F}infty
l{F}T
Additionally, then combining this above observation in this case, we see that the process
\tilde{W}t=Wt-\int
tY | |
sds |
for
t\in[0,T]
This theorem can be used to show in the Black–Scholes model the unique risk-neutral measure, i.e. the measure in which the fair value of a derivative is the discounted expected value, Q, is specified by
dQ | |
dP |
=l{E}\left(
t | |
\int | |
0 |
rs-\mus | |
\sigmas |
dWs\right).
Another application of this theorem, also given in the original paper of Igor Girsanov, is for stochastic differential equations. Specifically, let us consider the equation
dXt=\mu(t,Xt)dt+dWt,
where
Wt
\mu
\sigma
[0,T]
Xt
\Phi
C([0,T])
E\Phi(X)=E\left[
T | |
\Phi(W)\exp\left(\int | |
0 |
\mu(s,Ws)dW
|
2ds\right)\right]. | |
\int | |
s) |
This follows by applying Girsanov's theorem, and the above observation, to the martingale process
Yt=\int
t\mu(s,W | |
s)dW |
s.
In particular, we note that with the notation above, the process
\tilde{W}t=Wt-\int
t\mu(s,W | |
s)ds |
is a Q Brownian motion. Rewriting this in differential form as
dWt=d\tilde{W}t+\mu(t,Wt)dt,
we see that the law of
Wt
Xt
\tilde{W}t
EQ[\Phi(W)]
A more general form of this application is that if both
dXt=\mu(Xt,t)dt+\sigma(Xt,t)dWt,
dYt=(\mu(Yt,t)+\nu(Yt,t))dt+\sigma(Yt,t)dWt,
admit unique strong solutions on
[0,T]
C([0,T])
E\Phi(X)=E\left[
T | |
\Phi(Y)\exp\left(-\int | |
0 |
\nu(Ys,s) | |
\sigma(Ys,s) |
dW | ||||
|
T | |
\int | |
0 |
| |||||||
|
ds\right)\right].