Martingale representation theorem

Martingale representation theorem This article includes a list of general references, aber es fehlen genügend entsprechende Inline-Zitate. Bitte helfen Sie mit, diesen Artikel zu verbessern, indem Sie genauere Zitate einfügen. (Oktober 2011) (Erfahren Sie, wie und wann Sie diese Vorlagennachricht entfernen können) In der Wahrscheinlichkeitstheorie, 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, zum Beispiel, by Markov chains.

Statement Let {Anzeigestil B_{t}} be a Brownian motion on a standard filtered probability space {Anzeigestil (Omega ,{mathematisch {F}},{mathematisch {F}}_{t},P)} und lass {Anzeigestil {mathematisch {G}}_{t}} be the augmented filtration generated by {Anzeigestil B} . If X is a square integrable random variable measurable with respect to {Anzeigestil {mathematisch {G}}_{unendlich }} , then there exists a predictable process C which is adapted with respect to {Anzeigestil {mathematisch {G}}_{t}} , so dass {displaystyle X=E(X)+int _{0}^{unendlich }C_{s},dB_{s}.} Folglich, {Anzeigestil E(X|{mathematisch {G}}_{t})=E(X)+int _{0}^{t}C_{s},dB_{s}.} Application in finance The martingale representation theorem can be used to establish the existence of a hedging strategy. Nehme an, dass {Anzeigestil links(M_{t}Rechts)_{0leq t

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