Martingale representation theorem

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

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Let   be a Brownian motion on a standard filtered probability space   and let   be the augmented filtration generated by  . If X is a square integrable random variable measurable with respect to  , then there exists a predictable process C which is adapted with respect to   such that

 

Consequently,

 

Application in finance

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The martingale representation theorem can be used to establish the existence of a hedging strategy. Suppose that   is a Q-martingale process, whose volatility   is always non-zero. Then, if   is any other Q-martingale, there exists an  -previsible process  , unique up to sets of measure 0, such that   with probability one, and N can be written as:

 

The replicating strategy is defined to be:

  • hold   units of the stock at the time t, and
  • hold   units of the bond.

where   is the stock price discounted by the bond price to time   and   is the expected payoff of the option at time  .

At the expiration day T, the value of the portfolio is:

 

and it is 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  .

See also

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References

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  • Montin, Benoît. (2002) "Stochastic Processes Applied in Finance" [full citation needed]
  • Elliott, Robert (1976) "Stochastic Integrals for Martingales of a Jump Process with Partially Accessible Jump Times", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 36, 213–226