In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, the martingale representation theorem states that a random variable which is measurable with respect to the filtration generated by a Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

 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

Malliavin calculus

The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....

.

Similar theorems also exist for martingales

Martingale (probability theory)

In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

 on filtrations induced by jump processes, for example, by Markov chain

Markov chain

A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...

s.

Statement of the theorem

Let be a Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

 on a standard filtered probability space  and let be the augmentation of the filtration generated by . If X is a square integrable random variable measurable with respect to , then there exists a predictable process

Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process which the value is knowable at a prior time...

 C which is adapted

Adapted process

In the study of stochastic processes, an adapted process is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n...

with respect to , such that


Consequently

Application in finance

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 F-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'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 .