the martingale representation theorem
The martingale representation theorem is a completeness statement about Brownian motion: it says that in a Brownian world there are no martingales other than stochastic integrals against the Brownian motion. Every source of randomness, every fair-game process, is built out of Brownian increments and nothing else. This is the abstract reason markets driven by Brownian motion are complete (every contingent claim can be hedged), and it is the existence theorem behind backward SDEs and the explicit construction of hedging strategies.
Precisely: let B be a Brownian motion and let (F_t) be its augmented natural filtration (the filtration generated by B, satisfying the usual conditions). Then every square-integrable martingale M with respect to (F_t) has a representation M_t = M_0 + integral_0^t H_s dB_s for a unique (up to indistinguishability) predictable process H with E[integral_0^T H^2 ds] < infinity. Equivalently, every F_T-measurable square-integrable random variable Y can be written Y = E[Y] + integral_0^T H_s dB_s. The d-dimensional version replaces the single integral by a sum sum_i integral H^i dB^i. The deep content is that the Brownian filtration has the predictable representation property: the stochastic integrals against B already exhaust all martingales of the filtration. One clean proof route uses the density of stochastic exponentials (or Wiener-chaos / Hermite polynomials) in L^2 of the Brownian filtration, then represents each by Ito's formula and extends by linearity and L^2-density.
Its uses are foundational. In finance it is the mathematical content of market completeness: any payoff is the terminal value of a self-financing strategy integral H dS, so H is the hedge and the theorem guarantees it exists (though, honestly, it is an EXISTENCE theorem and usually non-constructive — finding H explicitly is the hard part, handled by Clark-Ocone or PDE methods). It is the existence backbone of backward stochastic differential equations and of filtering. The honest caveats that cannot be dropped: (1) The filtration must be the BROWNIAN one (its augmented natural filtration). If you enlarge the filtration with extra independent randomness, representation fails — there are then martingales (e.g. an independent compensated Poisson process) that are not Brownian integrals, and the market is incomplete. (2) The theorem gives existence and uniqueness of H but not, in general, a formula; the Clark-Ocone formula provides one under Malliavin-differentiability. (3) It is special to (Brownian and, more generally, Levy with the representation property) filtrations; a general filtration does NOT have the predictable representation property, which is precisely the abstract signature of an incomplete market.
Represent Y = B_T^2 (an F_T-measurable square-integrable variable) as a constant plus a Brownian integral. By Ito, B_t^2 = integral_0^t 2 B_s dB_s + t, so B_T^2 = T + integral_0^T 2 B_s dB_s = E[B_T^2] + integral_0^T 2 B_s dB_s. The integrand is H_s = 2 B_s and the constant E[Y] = T — exactly the representation the theorem promises, here found explicitly by Ito's formula.
Y = B_T^2 = E[Y] + integral H dB with H = 2B; in the Brownian filtration every L^2 variable has such a representation.
The theorem holds only for the Brownian filtration (or filtrations with the predictable representation property); enlarge it with independent jumps or extra randomness and representation fails — the abstract face of market incompleteness. It is also typically an existence result, not a recipe for the integrand H.