Girsanov theorem
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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.
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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.
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)_s</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>
If <math>Z_t</math> is a martingale then a probability
measure Q can be defined on <math>\{\sigma,F\}</math> such that 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 the process
- <math>\tilde W_t = W_t - <W,X>_t</math>
is a Wiener process on the filtered probability space <math>\{\Omega,F,Q,\{F^W_t\}\}</math>
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 Z to be a martingale is Novikov's condition which requires that
- <math> E_P\left [\exp\left (\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 Y_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 (for <math>\lambda\ne0</math>).
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^\cdot \frac{r_t - \mu_t }{\sigma_t}\,
d W \right )</math>
External links
- Notes on Stochastic Calculus which contain a simple outline proof of Girsanov's theorem.
References
- C. Dellacherie and P-A. Meyer, "Probabilities et potentiel -- Theorie de Martingales" Chapitre VII, Hermann 1980
- E. Lenglart "Transformation de martingales locales par changement absolue continu de probabilités", Zeitscrift für Wahrscheinlichkeit 39 (1977) pp 65-70.