Martingale representation theorem

Martingale representation theorem This article includes a list of general references, mais il manque suffisamment de citations en ligne correspondantes. Merci d'aider à améliorer cet article en introduisant des citations plus précises. (Octobre 2011) (Découvrez comment et quand supprimer ce modèle de message) En théorie des probabilités, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.

The theorem only asserts the existence of the representation and does not help to find it explicitly; it is possible in many cases to determine the form of the representation using Malliavin calculus.

Similar theorems also exist for martingales on filtrations induced by jump processes, par exemple, by Markov chains.

Déclaration Let {style d'affichage B_{t}} be a Brownian motion on a standard filtered probability space {style d'affichage (Oméga ,{mathématique {F}},{mathématique {F}}_{t},P)} et laissez {style d'affichage {mathématique {g}}_{t}} be the augmented filtration generated by {style d'affichage B} . If X is a square integrable random variable measurable with respect to {style d'affichage {mathématique {g}}_{infime }} , then there exists a predictable process C which is adapted with respect to {style d'affichage {mathématique {g}}_{t}} , tel que {displaystyle X=E(X)+entier _{0}^{infime }C_{s},dB_{s}.} Par conséquent, {style d'affichage E(X|{mathématique {g}}_{t})=E(X)+entier _{0}^{t}C_{s},dB_{s}.} Application in finance The martingale representation theorem can be used to establish the existence of a hedging strategy. Supposer que {style d'affichage à gauche(M_{t}droit)_{0leq t

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