Girsanov theorem

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.

History
Results of this type were first proved by Cameron-Martin in the 1940s and by Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977).

Significance
Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure absolutely continuous with respect to P then every P-semimartingale is a Q-semimartingale.

Statement of theorem
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 and in many other models (eg all continuous models).

Let $$\{W_t\}$$ be a Wiener process on the Wiener probability space $$\{\Omega,\mathcal{F},P\}$$. Let $$X_t$$ be a measurable process adapted to the natural filtration of the Wiener process $$\{\mathcal{F}^W_t\}$$.

Given an adapted process $$X_t$$ define


 * $$Z_t=\mathcal{E} (X)_t,\,$$

where $$\mathcal{E}(X)$$ is the stochastic exponential (or Doléans exponential) of X with respect to W, i.e.


 * $$\mathcal{E}(X)_t=\exp \left ( X_t - \frac{1}{2} [X]_t \right ).$$

If $$Z_t$$ is a martingale then a probability measure Q can be defined on $$\{\sigma,F\}$$ such that Radon-Nikodym derivative


 * $$\left .\frac{d Q}{d P} \right|_{\mathcal{F}_t} = Z_t = \mathcal{E} (X )_t$$

Then for each t the measure Q restricted to the unaugmented sigma fields $$\mathcal{F}^W_t$$ is equivalent to P restricted to $$\mathcal{F}^W_t.\,$$

Furthermore if Y is a local martingale under P then the process


 * $$\tilde Y_t = Y_t - \left[ W,X \right]_t$$

is a Q local martingale on the filtered probability space $$\{\Omega,F,Q,\{F^W_t\}\}$$.

Corollary
If X is a continuous process and W is Brownian Motion under measure P then
 * $$ \tilde W_t =W_t - \left [ W, X \right]_t $$

is Brownian motion under Q.

The fact that $$ \tilde W_t$$ is continuous is trivial; by Girsanov's theorem it is a Q local martingale, and by computing


 * $$\left[\tilde W \right]_t= \left [ W \right]_t = t$$

it follows by Levy's characterization of Brownian Motion that this is a Q Brownian Motion.

Comments
In many common applications, the process X is defined by


 * $$X_t = \int_0^t Y_s\, d W_s.$$

For X of this form then a necessary and sufficient condition for X to be a martingale is Novikov's condition which requires that


 * $$ E_P\left [\exp\left (\frac{1}{2}\int_0^T Y_s^2\, ds\right )\right ] < \infty. $$

The stochastic exponential $$\mathcal{E}(X)$$ is the process Z which solves the stochastic differential equation


 * $$ Z_t = 1 + \int_0^t Z_s\, d Y_s.\, $$

The measure Q constructed above is not equivalent to P on $$\mathcal{F}_\infty$$ as this would only be the case if the Radon-Nikodym derivative were a uniformly integrable martingale, which the exponential martingale described above is not (for $$\lambda\ne0$$).

Application to finance
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


 * $$ \frac{d Q}{d P} = \mathcal{E}\left ( \int_0^\cdot \frac{r_s - \mu_s }{\sigma_s}\,

d W_s \right ).$$