Girsanov theorem

In probability theory, the set of Girsanow is used to change stochastic processes. This happens using a Maßwechsels of the canonical measure P to the equivalent martingale Q. This set has a special meaning in financial mathematics, since under the equivalent martingale Martingale are the discounted prices of an underlying instrument, such as a stock. In the field of stochastic processes of Maßwechsel is important because then the following statement can be made: If Q is a probability measure absolutely continuous with respect to P, then each P - semimartingale is a Q- semimartingale.

History

The theorem was first proved in 1945 by Cameron and Martin and then in 1960 by Igor Vladimirovich Girsanow. The theorem was generalized by Lenglart 1977.

Set

Let be a probability space, equipped with the natural filtration of the standard Wiener process. Be an adapted process, such that P- almost - surely and the process defined by

Is a martingale.

Then under the probability measure with density with respect, that the process is defined by a standard Wiener process.

Comments

The process is the stochastic exponential of the process, that is, it solves the stochastic differential equation. He is always a non-negative local martingale, so also a supermartingale. The generally difficult part in the application of the above theorem is the condition that is actually a martingale. A sufficient condition so that a martingale is:

This condition is also called the Novikov condition.

Credentials

• C. Dellacherie, P.-A. Meyer, " probabilites et potentiel - Théorie des martingales " Chapter VII, Hermann 1980
• Damien Lamberton and Bernard Lapeyre, "Introduction to Stochastic Calculus Applied to Finance", Chapter IV, p 66, Chapman & Hall, 2000, ISBN 0-412-71800-6