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| [[Image:Girsanov.png|thumb|400px|Visualisation of the Girsanov theorem — The left side shows a [[Wiener process]] with negative drift under a canonical measure ''P''; on the right side each path of the process is colored according to its [[likelihood]] under the [[martingale (probability theory)|martingale]] measure ''Q''. The density transformation from ''P'' to ''Q'' is given by the Girsanov theorem.]]
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| In [[probability theory]], the '''Girsanov theorem''' (named after [[Igor Vladimirovich Girsanov]]) describes how the dynamics of [[stochastic process]]es change when the original [[measure (probability)|measure]] is changed to an [[Equivalence (measure theory)|equivalent probability measure]]<ref>M. Musiela, M. Rutkowski: Martingale methods in financial modelling. 2nd ed. New York : Springer-Verlag, 2004. Print.</ref>{{rp|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 [[stock|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 [[derivative (finance)|derivatives]] on the underlying.
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| ==History==
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| 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).
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| ==Significance==
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| Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if ''Q'' is a [[measure (mathematics)|measure]] [[absolute continuity|absolutely continuous]] with respect to ''P'' then every ''P''-[[semimartingale]] is a ''Q''-semimartingale.
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| ==Statement of theorem==
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| 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).
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| 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 process|adapted]] to the [[natural filtration|natural]] [[filtration (abstract algebra)|filtration]] of the Wiener process <math>\{\mathcal{F}^W_t\}</math>.
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| Given an adapted process <math>X_t</math> with <math>X_0 = 0</math> define
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| :<math>Z_t=\mathcal{E} (X)_t,\,</math>
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| where <math>\mathcal{E}(X)</math> is the stochastic exponential (or [[Doléans exponential]]) of ''X'' with respect to ''W'', i.e.
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| :<math>\mathcal{E}(X)_t=\exp \left ( X_t - \frac{1}{2} [X]_t \right ),</math>
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| where <math> [X]_t </math> is a [[quadratic variation]] for <math>X_t</math>. Thus <math>Z_t</math> is a strictly positive [[local martingale]], and a probability
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| measure ''Q'' can be defined on <math>\{\Omega,\mathcal{F}\}</math> such that we have [[Radon–Nikodym derivative]]
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| :<math>\frac{d Q}{d P} |_{\mathcal{F}_t} = Z_t = \mathcal{E} (X )_t</math>
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| Then for each ''t'' the measure ''Q'' restricted to the unaugmented sigma fields <math>\mathcal{F}^W_t</math> is equivalent to ''P'' restricted to <math>\mathcal{F}^W_t.\,</math>
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| Furthermore if ''Y'' is a local martingale under ''P'' then the process
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| :<math>\tilde Y_t = Y_t - \left[ Y,X \right]_t</math>
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| is a ''Q'' local martingale on the [[filtered probability space]] <math>\{\Omega,F,Q,\{F^W_t\}\}</math>.
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| ==Corollary==
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| If ''X'' is a continuous process and ''W'' is [[Wiener process|Brownian motion]] under measure ''P'' then
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| :<math> \tilde W_t =W_t - \left [ W, X \right]_t </math>
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| is Brownian motion under ''Q''.
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| The fact that <math> \tilde W_t</math> is continuous is trivial; by Girsanov's theorem it is a ''Q'' local martingale, and by computing the [[quadratic variation]]
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| :<math>\left[\tilde W \right]_t= \left [ W \right]_t = t</math>
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| it follows by [[Wiener_process#Characterizations of the Wiener process|Lévy's characterization]] of Brownian motion that this is a ''Q'' Brownian
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| motion.
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| ===Comments===
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| In many common applications, the process ''X'' is defined by
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| :<math>X_t = \int_0^t Y_s\, d W_s.</math>
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| For ''X'' of this form then a sufficient condition for <math>\mathcal{E}(X)</math> to be a martingale is [[Novikov's condition]] which requires that
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| :<math> E_P\left [\exp\left (\frac{1}{2}\int_0^T Y_s^2\, ds\right )\right ] < \infty. </math>
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| The stochastic exponential <math>\mathcal{E}(X)</math> is the process ''Z'' which solves the stochastic differential equation
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| :<math> Z_t = 1 + \int_0^t Z_s\, d X_s.\, </math>
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| 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 theorem|Radon–Nikodym derivative]] were a uniformly integrable martingale, which the exponential martingale described above is not (for <math>\lambda\ne0</math>).
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| ==Application to finance==
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| 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 framework - via [[Radon–Nikodym theorem|Radon–Nikodym derivative]]:
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| <math> \frac{d Q}{d P} = \mathcal{E}\left ( \int_0^\cdot \frac{r - \mu }{\sigma}\,
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| d W_s \right )</math>
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| where <math> r </math> denotes the instanteaneous risk free rate, <math>\mu</math> the asset's drift and <math>\sigma</math> its volatility.
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| Other classical applications of Girsanov theorem are quanto adjustments and the calculation of forwards' drifts under [[LIBOR market model]].
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| ==See also==
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| *[[Cameron–Martin theorem]]
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| ==References==
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| * C. Dellacherie and P.-A. Meyer, "Probabilités et potentiel -- Théorie de Martingales" Chapitre VII, Hermann 1980
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| * Girsanov, I. V., "On transforming a certain class of stochastic processes by absolutely continuous substitution of measures", Theory of Probability and its Applications, 1960
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| * E. Lenglart "Transformation de martingales locales par changement absolue continu de probabilités", Zeitschrift für Wahrscheinlichkeit 39 (1977) pp 65–70.
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| ==External links==
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| {{reflist}}
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| * [http://www.chiark.greenend.org.uk/~alanb/stoc-calc.pdf Notes on Stochastic Calculus] which contains a simple outline proof of Girsanov's theorem.
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| * [http://ssrn.com/abstract=1805984 Applied Multidimensional Girsanov Theorem] which contains financial applications of Girsanov's theorem.
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| {{DEFAULTSORT:Girsanov Theorem}}
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| [[Category:Stochastic processes]]
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| [[Category:Probability theorems]]
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