Editing Girsanov theorem

{{Short description|Theorem on changes in stochastic processes}}
[[Image:Girsanov.png|thumb|400px|Visualisation of the Girsanov theorem. The left side shows a [[Wiener process]] with negative drift under a canonical measure ''P''; on the right side each path of the process is colored according to its [[likelihood]] under the martingale measure ''Q''. The density transformation from ''P''  to ''Q'' is given by the Girsanov theorem.]]
In [[probability theory]],  '''Girsanov's theorem''' or the '''Cameron-Martin-Girsanov theorem''' explains how [[stochastic process]]es change under changes in [[measure (probability)|measure]]. The theorem is especially important in the theory of [[financial mathematics]] as it explains how to convert from the [[physical measure]], which describes the probability that an [[underlying instrument]] (such as a [[stock|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 [[derivative (finance)|derivatives]] on the underlying.

==History==
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).

==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 (mathematics)|measure]] that is [[absolute continuity|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]].

Let <math>\{W_t\}</math> be a Wiener process on the Wiener probability space <math>\{\Omega,\mathcal{F},P\}</math>. Let <math>X_t</math> be a measurable process adapted to the natural filtration of the Wiener process <math>\{\mathcal{F}^W_t\}</math>; we assume that the usual conditions have been satisfied.

Given an adapted process <math>X_t</math> define

:<math>Z_t=\mathcal{E} (X)_t,\,</math>

where <math>\mathcal{E}(X)</math> is the [[stochastic exponential]] of ''X'' with respect to ''W'', i.e.

:<math>\mathcal{E}(X)_t=\exp \left ( X_t - \frac{1}{2} [X]_t \right ),</math>

and <math>[X]_t</math> denotes the [[quadratic variation]] of the process ''X''.

If <math>Z_t</math> is a [[martingale (probability)|martingale]] then a probability 
measure ''Q'' can be defined on <math>\{\Omega,\mathcal{F}\}</math> such that [[Radon–Nikodym derivative]]

:<math>\left .\frac{d Q}{d P} \right|_{\mathcal{F}_t} = Z_t = \mathcal{E} (X )_t</math>

Then for each ''t'' the measure ''Q'' restricted to the unaugmented sigma fields <math>\mathcal{F}^o_t</math> is equivalent to ''P'' restricted to

:<math>\mathcal{F}^o_t.\,</math>

Furthermore, if <math>Y_t</math> is a local martingale under ''P'' then the process

:<math>\tilde Y_t = Y_t - \left[ Y,X \right]_t</math>

is a ''Q'' local martingale on the filtered probability space <math>\{\Omega,F,Q,\{\mathcal{F}^W_t\}\}</math>.

==Corollary==
If ''X'' is a continuous process and ''W'' is a Brownian motion under measure ''P'' then
:<math> \tilde W_t =W_t -  \left [ W, X \right]_t </math>
is a Brownian motion under ''Q''.

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

:<math>\left[\tilde W \right]_t= \left [ W \right]_t = t</math>

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

:<math>X_t = \int_0^t Y_s\, d W_s.</math>

For ''X'' of this form then a necessary and sufficient condition for <math>\mathcal{E}(X)</math> to be a martingale is [[Novikov's condition]] which requires that

:<math> E_P\left [\exp\left (\frac{1}{2}\int_0^T Y_s^2\, ds\right )\right ] < \infty. </math>

The stochastic exponential <math>\mathcal{E}(X)</math> is the process ''Z'' which solves the stochastic differential equation

:<math> Z_t = 1 + \int_0^t Z_s\, d X_s.\, </math>

The measure ''Q'' constructed above is not equivalent to ''P'' on <math>\mathcal{F}_\infty</math> 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. On the other hand, as long as Novikov's condition is satisfied the measures are equivalent on <math> \mathcal{F}_T </math>.

Additionally, then combining this above observation in this case, we see that the process

<math> \tilde{W}_t=W_t-\int_0^tY_sds </math>

for <math> t\in [0,T] </math> is a Q Brownian motion. This was Igor Girsanov's original formulation of the above theorem.

==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

:<math> \frac{d Q}{d P} = \mathcal{E}\left ( \int_0^t \frac{r_s - \mu_s }{\sigma_s}\,
d W_s \right ).</math>

== Application to Langevin equations ==
Another application of this theorem, also given in the original paper of Igor Girsanov, is for [[stochastic differential equation]]s. Specifically, let us consider the equation

<math> dX_t=\mu(t, X_t)dt+dW_t, </math>

where <math> W_t </math> denotes a Brownian motion. Here <math> \mu </math> and <math> \sigma </math> are fixed deterministic functions. We assume that this equation has a unique strong solution on <math> [0,T] </math>. In this case Girsanov's theorem may be used to compute functionals of <math> X_t </math> directly in terms a related functional for Brownian motion. More specifically, we have for any bounded functional <math> \Phi </math> on continuous functions <math> C([0,T]) </math> that

<math> E \Phi(X)=E\left[ \Phi(W)\exp\left(\int_0^T \mu(s,W_s)dW_s-\frac{1}{2}\int_0^T\mu(s,W_s)^2ds\right)\right]. </math>

This follows by applying Girsanov's theorem, and the above observation, to the martingale process

<math> Y_t=\int_0^t\mu(s,W_s)dW_s. </math>

In particular, with the notation above, the process

<math> \tilde{W}_t=W_t-\int_0^t\mu(s,W_s)ds </math>

is a Q Brownian motion. Rewriting this in differential form as

<math> dW_t=d\tilde{W}_t+\mu(t,W_t)dt, </math>

we see that the law of <math> W_t </math> under Q solves the equation defining <math> X_t </math>, as <math> \tilde{W}_t </math> is a Q Brownian motion. In particular, we see that the right-hand side may be written as <math> E_Q[\Phi(W)] </math>, where Q is the measure taken with respect to the process Y, so the result now is just the statement of Girsanov's theorem.

A more general form of this application is that if both

<math> dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dW_t, </math> <math> dY_t=(\mu(Y_t,t)+\nu(Y_t,t))dt+\sigma(Y_t,t)dW_t, </math>

admit unique strong solutions on <math> [0,T] </math>, then for any bounded functional on  <math> C([0,T]) </math>, we have that

<math> E \Phi(X)=E\left[ \Phi(Y)\exp\left(-\int_0^T \frac{\nu(Y_s,s)}{\sigma(Y_s,s)}dW_s-\frac{1}{2}\int_0^T \frac{\nu(Y_s,s)^2}{\sigma(Y_s,s)^2}ds\right)\right]. </math>

==See also==

* {{annotated link|Cameron–Martin theorem}}

==References==

{{reflist}}

* {{cite book |last=Liptser |first=Robert S. |last2=Shiriaev |first2=A. N. |title=Statistics of Random Processes |publisher=Springer |edition=2nd, rev. and exp. |year=2001 |isbn=3-540-63929-2 }}
* {{cite book |first=C. |last=Dellacherie |first2=P.-A. |last2=Meyer |translator-last=Wilson |translator-first=J. P. |title=Probabilities and Potential |volume=B |chapter=Decomposition of Supermartingales, Applications |pages=183–308 |publisher=North-Holland |year=1982 |isbn=0-444-86526-8 }}
* {{cite journal |first=E. |last=Lenglart |title=Transformation de martingales locales par changement absolue continu de probabilités |language=fr |journal=Zeitschrift für Wahrscheinlichkeit |volume=39 |issue= |year=1977 |pages=65–70 |doi=10.1007/BF01844873 |doi-access=free }}

==External links==
* [http://www.chiark.greenend.org.uk/~alanb/stoc-calc.pdf Notes on Stochastic Calculus] which contain a simple outline proof of Girsanov's theorem.

[[Category:Stochastic processes]]
[[Category:Mathematical theorems]]
[[Category:Mathematical finance]]