# Girsanov theorem

In probability theory, the **Girsanov theorem** (named after Igor Vladimirovich Girsanov) describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure.[1]^{:607} 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 pricing derivatives on the underlying instrument.

## History

Results of this type were first proved by Cameron–Martin in the 1940s and by Girsanov in 1960.[2] They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977).[3]

## 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 (e.g. all continuous models).

Let be a Wiener process on the Wiener probability space . Let be a measurable process adapted to the natural filtration of the Wiener process with .

Define the Doléans-Dade exponential of *X* with respect to *W*

where is the quadratic variation of . If is a strictly positive martingale, a probability
measure *Q* can be defined on such that we have Radon–Nikodym derivative

Then for each *t* the measure *Q* restricted to the unaugmented sigma fields is equivalent to *P* restricted to . Furthermore, if *Y* is a local martingale under *P*, then the process

is a *Q* local martingale on the filtered probability space .

## Corollary

If *X* is a continuous process and *W* is Brownian motion under measure *P* then

is Brownian motion under *Q*.

The fact that is continuous is trivial; by Girsanov's theorem it is a *Q* local martingale, and by computing the quadratic variation

it follows by Lévy's characterization of Brownian motion that this is a *Q* Brownian
motion.

### Comments

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

If *X* is of this form, then a sufficient condition for to be a martingale is Novikov's condition, which requires that

The stochastic exponential is the process *Z*, which solves the stochastic differential equation

The measure *Q* constructed above is not equivalent to *P* on , 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 ).

## Application to finance

In finance, Girsanov theorem is used each time one needs to derive an asset's or rate's dynamics under a new probability measure. The most well known case is moving from historic measure P to risk neutral measure Q which is done—in Black–Scholes model—via Radon–Nikodym derivative:

where denotes the instantaneous risk free rate, the asset's drift and its volatility.

Other classical applications of Girsanov theorem are quanto adjustments and the calculation of forwards' drifts under LIBOR market model.

## See also

## References

- Musiela, M.; Rutkowski, M. (2004).
*Martingale Methods in Financial Modelling*(2nd ed.). New York: Springer. ISBN 3-540-20966-2. - Girsanov, I. V. (1960). "On transforming a certain class of stochastic processes by absolutely continuous substitution of measures".
*Theory of Probability and Its Applications*.**5**(3): 285–301. doi:10.1137/1105027. - Lenglart, É. (1977). "Transformation des martingales locales par changement absolument continu de probabilités".
*Zeitschrift für Wahrscheinlichkeit*.**39**(1): 65–70. doi:10.1007/BF01844873.

- Dellacherie, C.; Meyer, P.-A. (1980).
*Probabilités et potentiel: Théorie de Martingales: Chapitre VII*(in French). Paris: Hermann. ISBN 2-7056-1385-4.

## External links

- Notes on Stochastic Calculus which contains a simple outline proof of Girsanov's theorem.
- Papaioannou, Denis (July 14, 2012). "Applied Multidimensional Girsanov Theorem". SSRN 1805984. Cite journal requires
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(help) Contains financial applications of Girsanov's theorem.