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The martingale representation theorem states that every square-integrable martingale M(t) adapted to the natural filtration of a Brownian motion W can be written as M(t) = M(0) + ∫₀ᵗ H(s) dW(s) for some adapted process H. In other words, Brownian motion is the only source of randomness in its own filtration — every martingale is an Itô integral against W. This result underpins the completeness of the Black-Scholes market and the existence of perfect hedging strategies.
The martingale representation theorem reveals a striking structural property of the Brownian filtration: every source of randomness in the system is already captured by the Brownian motion itself. Formally, if M(t) is any square-integrable martingale adapted to the natural filtration ℱ_tW of a Brownian motion W, then there exists an adapted, square-integrable process H(t) such that M(t) = M(0) + ∫₀ᵗ H(s) dW(s). No martingale in this filtration is "orthogonal" to W — everything is an Itô integral.
The proof relies on the fact that the exponential martingales Z_θ(t) = exp(θW(t) - θ²t/2) span L²(ℱ_TW) as θ varies over the reals. Since each Z_θ is itself an Itô integral (dZ_θ = θZ_θ dW), any L² random variable measurable with respect to ℱ_TW can be approximated by sums of Itô integrals, and hence is itself an Itô integral. The process H in the representation M(t) = M(0) + ∫₀ᵗ H dW is the integrand that replicates M using W — in financial terms, it is the hedging strategy.
The most important application is to market completeness in mathematical finance. In the Black-Scholes model, the stock price S follows geometric Brownian motion under the risk-neutral measure Q, and the filtration is generated by the driving Brownian motion. Any contingent claim with payoff C(S_T) at maturity has a price process V(t) = E_Q[e-r(T-t)C(S_T) | ℱ_t] that is a Q-martingale (after discounting). By the martingale representation theorem, V(t) = V(0) + ∫₀ᵗ H(s) dW̃(s) for some H. Converting back to stock-denominated units gives the replicating portfolio: hold H(t)/σS(t) shares of stock at time t, and the portfolio exactly replicates C(S_T) at maturity. This is why Black-Scholes hedging works — the theorem guarantees the existence of a perfect hedge.
The theorem fails when the filtration contains more randomness than a single Brownian motion can generate. In stochastic volatility models (where volatility itself is random), the filtration is generated by two Brownian motions, and the representation requires two integrals: M = M(0) + ∫H₁ dW₁ + ∫H₂ dW₂. With only one tradeable asset (the stock, driven by W₁), you cannot replicate claims that depend on W₂ — the market is incomplete, and perfect hedging is impossible. The number of independent Brownian motions in the filtration determines the number of hedging instruments needed for completeness.