Girsanov theorem

In probability theory, Girsanov's theorem or the Cameron-Martin-Girsanov theorem explains how stochastic processes change under changes in 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 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.

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).

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

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.

; we assume that the usual conditions have been satisfied.

denotes the quadratic variation of the process X.

is a martingale then a probability measure Q can be defined on

such that Radon–Nikodym derivative Then for each t the measure Q restricted to the unaugmented sigma fields

is a local martingale under P then the process is a Q local martingale on the filtered probability space

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

is continuous is trivial; by Girsanov's theorem it is a Q local martingale, and by computing it follows by Levy's characterization of Brownian motion that this is a Q Brownian motion.

In many common applications, the process X is defined by For X of this form then a necessary and 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.

On the other hand, as long as Novikov's condition is satisfied the measures are equivalent on

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

This was Igor Girsanov's original formulation of the above theorem.

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 Another application of this theorem, also given in the original paper of Igor Girsanov, is for stochastic differential equations.

We assume that this equation has a unique strong solution on

In this case Girsanov's theorem may be used to compute functionals of

directly in terms a related functional for Brownian motion.

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

Rewriting this in differential form as

under Q solves the equation defining

, where Q is the measure taken with respect to the process Y, so the result now is just the statement of Girsanov's theorem.

admit unique strong solutions on