Inprobability theory,Girsanov's theorem or theCameron-Martin-Girsanov theorem explains howstochastic processes change under changes inmeasure. The theorem is especially important in the theory offinancial mathematics as it explains how to convert from thephysical measure, which describes the probability that anunderlying instrument (such as ashare price orinterest rate) will take a particular value or values, to therisk-neutral measure which is a very useful tool for evaluating the value ofderivatives on the underlying.

Results of this type were first proved by Cameron-Martin in the 1940s and byIgor 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 ifQ is ameasure that isabsolutely continuous with respect toP then everyP-semimartingale is aQ-semimartingale.

We state the theorem first for the special case when the underlying stochastic process is aWiener process. This special case is sufficient for risk-neutral pricing in theBlack–Scholes model.

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

Given an adapted process${\displaystyle X_t}$ define

${\displaystyle Z_t=\mathcal{E}(X{)}_t,}$

where${\displaystyle \mathcal{E}(X)}$ is thestochastic exponential ofX with respect toW, i.e.

${\displaystyle \mathcal{E}(X{)}_t=\exp \left(X_t-\frac{1}{2}[X{]}_t\right),}$

and${\displaystyle [X{]}_t}$ denotes thequadratic variation of the processX.

If${\displaystyle Z_t}$ is amartingale then a probability measureQ can be defined on${\displaystyle \{\mathrm{\Omega },\mathcal{F}\}}$ such thatRadon–Nikodym derivative

${\displaystyle {\frac{dQ}{dP}|}_{{\mathcal{F}}_t}=Z_t=\mathcal{E}(X{)}_t}$

Then for eacht the measureQ restricted to the unaugmented sigma fields${\displaystyle {\mathcal{F}}_t^o}$ is equivalent toP restricted to

${\displaystyle {\mathcal{F}}_t^o.}$

Furthermore, if${\displaystyle Y_t}$ is alocal martingale underP then the process

${\displaystyle {\tilde{Y}}_t=Y_t-{\left[Y,X\right]}_t}$

is aQ local martingale on the filtered probability space${\displaystyle \{\mathrm{\Omega },F,Q,\{{\mathcal{F}}_t^W\}\}}$.

IfX is a continuous process andW is a Brownian motion under measureP then

${\displaystyle {\tilde{W}}_t=W_t-{\left[W,X\right]}_t}$

is a Brownian motion underQ.

The fact that${\displaystyle {\tilde{W}}_t}$ is continuous is trivial; by Girsanov's theorem it is aQ local martingale, and by computing

${\displaystyle {\left[\tilde{W}\right]}_t={\left[W\right]}_t=t}$

it follows by Levy's characterization of Brownian motion that this is aQ Brownianmotion.

In many common applications, the processX is defined by

${\displaystyle X_t={\int }_0^t{Y}_{s}d{W}_s.}$

ForX of this form then a necessary and sufficient condition for${\displaystyle \mathcal{E}(X)}$ to be a martingale isNovikov's condition which requires that

The stochastic exponential${\displaystyle \mathcal{E}(X)}$ is the processZ which solves the stochastic differential equation

${\displaystyle Z_t=1+{\int }_0^t{Z}_{s}d{X}_s.}$

The measureQ constructed above is not equivalent toP on${\displaystyle {\mathcal{F}}_{\mathrm{\infty }}}$ 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${\displaystyle {\mathcal{F}}_T}$.

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

${\displaystyle {\tilde{W}}_t=W_t-{\int }_0^t{Y}_{s}ds}$

for${\displaystyle t\in [0,T]}$ is a Q Brownian motion. 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 discountedexpected value, Q, is specified by

${\displaystyle \frac{dQ}{dP}=\mathcal{E}\left({\int }_0^t\frac{r_s-{\mu }_s}{{\sigma }_s}d{W}_s\right).}$

Another application of this theorem, also given in the original paper of Igor Girsanov, is forstochastic differential equations. Specifically, let us consider the equation

${\displaystyle d{X}_t=\mu (X_t,t)dt+\sigma (X_t,t)d{W}_t,}$

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

${\displaystyle E\mathrm{\Phi }(X)=E\left[\mathrm{\Phi }(W)\exp \left({\int }_0^T\mu (W_s,s)d{W}_s-\frac{1}{2}{\int }_0^T\mu (W_s,s{)}^{2}ds\right)\right].}$

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

${\displaystyle Y_t={\int }_0^t\mu (W_s,s)d{W}_s.}$

In particular, with the notation above, the process

${\displaystyle {\tilde{W}}_t=W_t-{\int }_0^t\mu (W_s,s)ds}$

is a Q Brownian motion. Rewriting this indifferential form as

${\displaystyle d{W}_t=d{\tilde{W}}_t+\mu (W_t,t)dt,}$

we see that the law of${\displaystyle W_t}$ under Q solves the equation defining${\displaystyle X_t}$, as${\displaystyle {\tilde{W}}_t}$ is a Q Brownian motion. In particular, we see that the right-hand side may be written as${\displaystyle E_Q[\mathrm{\Phi }(W)]}$, 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

${\displaystyle d{X}_t=\mu (X_t,t)dt+\sigma (X_t,t)d{W}_t,}$${\displaystyle d{Y}_t=(\mu (Y_t,t)+\nu (Y_t,t))dt+\sigma (Y_t,t)d{W}_t,}$

admit unique strong solutions on${\displaystyle [0,T]}$, then for any bounded functional on${\displaystyle C([0,T])}$, we have that

${\displaystyle E\mathrm{\Phi }(X)=E\left[\mathrm{\Phi }(Y)\exp \left(-{\int }_0^T\frac{\nu (Y_s,s)}{\sigma (Y_s,s)}d{W}_s-\frac{1}{2}{\int }_0^T\frac{\nu (Y_s,s{)}^2}{\sigma (Y_s,s{)}^2}ds\right)\right].}$

• Cameron–Martin theorem – Theorem describing translation of Gaussian measures on Hilbert spaces It is a special case of the Girsanov theorem who has inspired all the theory.
• Girsanov theorem has fundamental applications to the Quantum Field Theory, Malliavin Calculus, stochastic partial differential equations, degree theory, calculus of variations on the classical and abstract Wiener spaces.

• Liptser, Robert S.; Shiriaev, A. N. (2001).Statistics of Random Processes (2nd, rev. and exp. ed.). Springer.ISBN 3-540-63929-2.
• Dellacherie, C.; Meyer, P.-A. (1982). "Decomposition of Supermartingales, Applications".Probabilities and Potential. Vol. B. Translated by Wilson, J. P. North-Holland. pp. 183–308.ISBN 0-444-86526-8.
• Lenglart, E. (1977)."Transformation de martingales locales par changement absolue continu de probabilités".Zeitschrift für Wahrscheinlichkeit (in French).39:65–70.doi:10.1007/BF01844873.
• \"Ust\"unel, A. S.; Zakai, M. (2000).Transformation of Measure on Wiener space (1st ed.). Springer.ISBN 3-540-66455-6.

• Notes on Stochastic Calculus which contain a simple outline proof of Girsanov's theorem.

• Stochastic processes
• Mathematical theorems
• Mathematical finance

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• CS1 French-language sources (fr)