Martingale representation theorem

Martingale representation theorem This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (October 2011) (Learn how and when to remove this template message) In probability theory, 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, for example, by Markov chains.

Statement Let {displaystyle B_{t}} be a Brownian motion on a standard filtered probability space {displaystyle (Omega ,{mathcal {F}},{mathcal {F}}_{t},P)} and let {displaystyle {mathcal {G}}_{t}} be the augmented filtration generated by {displaystyle B} . If X is a square integrable random variable measurable with respect to {displaystyle {mathcal {G}}_{infty }} , then there exists a predictable process C which is adapted with respect to {displaystyle {mathcal {G}}_{t}} , such that {displaystyle X=E(X)+int _{0}^{infty }C_{s},dB_{s}.} Consequently, {displaystyle E(X|{mathcal {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. Suppose that {displaystyle left(M_{t}right)_{0leq t

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