Martingale representation theorem

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In probability theory, the martingale representation theorem shows that a random variable which 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.

[edit] Statement of the theorem

Let $B t$ be a Brownian motion on a standard filtered probability space $(\Omega, \mathcal{F},\mathcal{F}_t, P )$ , and let X be a square integrable random variable measurable with respect to $\mathcal{F}_\infty$ . Then there exists a previsible process C which is adapted with respect to $\mathcal{F}_t$ , such that

$E(X| \mathcal{F}_t) = E(X) + \int_0^t C_s\,dB_s$ .

[edit] Applied in finance

The martingale representation theorem can be used to establish the existence of a hedging strategy. Suppose that $\left ( M_t \right )_{0 \le t < \infty}$ is a Q-martingale process, whose volatility $σ t$ is always non-zero. Then, if $\left ( N_t \right )_{0 \le t < \infty}$ is any other Q-martingale, there exists an F-previsible process $φ$ , unique up to sets of measure 0, such that $\int_0^T \phi_t^2 \sigma_t^2 \, dt < \infty$ with probability one, and N can be written as:

$N_t = N_0 + \int_0^t \phi_s\, d M_s$

The replicating strategy is defined to be:

hold $φ t$ units of the stock at the time t, and
hold $ψ t = C t - φ t Z t$ units of the bond.

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

V T = φ T S T + ψ T B T = B T C T = X

and it's easy to check that the strategy is self-financing: the change in the value of the portfolio only depends in the change of the asset prices $\left ( dV_t = \phi_t d S_t + \psi_t\, d B_t \right )$ .