Martingale representation theorem

In probability theory, the martingale representation theorem states 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.

Similar theorems also exist for martingales on filtrations induced by jump processes, for example, by Markov chains.

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 $\mathcal{G}_t$ be the augmentation of the filtration generated by B. If X is a square integrable random variable measurable with respect to $\mathcal{G}_\infty$, then there exists a predictable process C which is adapted with respect to $\mathcal{G}_t$, such that

$X = E(X) + \int_0^\infty C_s\,dB_s.$

Consequently

$E(X| \mathcal{G}_t) = E(X) + \int_0^t C_s \, d B_s.$

Application 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_tZ_t units of the bond.

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

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

V_T = ϕ_TS_T + ψ_TB_T = B_TC_T = X

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 $\left ( dV_t = \phi_t d S_t + \psi_t\, d B_t \right )$.

References

• 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 fuer Wahrscheinlichkeitstheorie und verwandte Gebiete, 36, 213-226