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| In [[probability theory]], the '''martingale representation theorem''' states that a random variable that is measurable with respect to the [[Filtration (mathematics)#Measure theory|filtration]] generated by a [[Brownian motion]] can be written in terms of an [[Itô integral]] with respect to this Brownian motion.
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| 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]].
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| Similar theorems also exist for [[Martingale (probability theory)|martingales]] on filtrations induced by jump processes, for example, by [[Markov chain]]s.
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| ==Statement of the theorem==
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| Let <math>B_t</math> be a [[Brownian motion]] on a standard [[filtered probability space]] <math>(\Omega, \mathcal{F},\mathcal{F}_t, P )</math> and let <math>\mathcal{G}_t</math> be the [[augmentation of the filtration]] generated by <math>B</math>. If ''X'' is a square integrable random variable measurable with respect to <math>\mathcal{G}_\infty</math>, then there exists a [[predictable process]] ''C'' which is [[adapted process|adapted]] with respect to <math>\mathcal{G}_t</math>, such that
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| :<math>X = E(X) + \int_0^\infty C_s\,dB_s.</math>
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| Consequently
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| :<math> E(X| \mathcal{G}_t) = E(X) + \int_0^t C_s \, d B_s.</math>
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| ==Application in finance==
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| The martingale representation theorem can be used to establish the existence
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| of a hedging strategy.
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| Suppose that <math>\left ( M_t \right )_{0 \le t < \infty}</math> is a Q-martingale process, whose volatility <math>\sigma_t</math> is always non-zero.
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| Then, if <math>\left ( N_t \right )_{0 \le t < \infty}</math> is any other Q-martingale, there exists an <math>\mathcal{F}</math>-previsible process <math>\phi</math>, unique up to sets of measure 0, such that <math>\int_0^T \phi_t^2 \sigma_t^2 \, dt < \infty</math> with probability one, and ''N'' can be written as:
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| :<math>N_t = N_0 + \int_0^t \phi_s\, d M_s.</math>
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| The replicating strategy is defined to be:
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| * hold <math>\phi_t</math> units of the stock at the time ''t'', and
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| * hold <math>\psi_t B_t = C_t - \phi_t Z_t</math> units of the bond.
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| where <math>Z_t</math> is the stock price discounted by the bond price to time <math>t</math> and <math>C_t</math> is the expected payoff of the option at time <math>t</math>.
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| At the expiration day ''T'', the value of the portfolio is:
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| :<math>V_T = \phi_T S_T + \psi_T B_T = C_T = X</math>
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| and it's easy to check that the strategy is self-financing: the change in the value of the portfolio only depends on the change of the asset prices <math>\left ( dV_t = \phi_t d S_t + \psi_t\, d B_t \right ) </math>.
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| {{inline|date=October 2011}}
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| ==References==
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| *Montin, Benoît. (2002) "Stochastic Processes Applied in Finance" {{full|date=November 2012}}
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| *[[Robert J. Elliott|Elliott, Robert]] (1976) "Stochastic Integrals for Martingales of a Jump Process with Partially Accessible Jump Times", ''Zeitschrift fuer Wahrscheinlichkeitstheorie und verwandte Gebiete'', 36, 213-226
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| [[Category:Martingale theory]]
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| [[Category:Probability theorems]]
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