Levy's characterization of Brownian motion
/ lev-EE (Lévy) /
Levy's characterization is a surprising and powerful identification theorem: it says you can recognize Brownian motion purely from two abstract martingale properties, without ever checking that increments are Gaussian and independent. This is what makes Brownian motion robust under stochastic-calculus operations — if a transformation produces a continuous martingale with the right quadratic variation, it must be a Brownian motion, no matter how it was built. It is the converse direction to '(dB)^2 = dt' and a cornerstone proof technique.
The theorem states: if M is a continuous local martingale with M_0 = 0 and quadratic variation [M]_t = t for all t, then M is a standard Brownian motion. The multidimensional version: if M = (M^1, ..., M^d) is a vector of continuous local martingales with M^i_0 = 0 and quadratic covariations [M^i, M^j]_t = delta_{ij} t (each component has clock t, distinct components are orthogonal), then M is a d-dimensional standard Brownian motion. The proof is a beautiful one-line application of Ito's formula: fix theta in R^d and apply Ito to the complex exponential exp(i theta . M_t + (1/2) |theta|^2 t); the (dB)^2 = dt correction exactly cancels the time drift, showing this is a local martingale, hence (after a uniform-integrability check) a true martingale, which forces E[exp(i theta . (M_t - M_s)) | F_s] = exp(-(1/2)|theta|^2 (t - s)). That is precisely the statement that increments are independent of the past and N(0, (t-s) I)-distributed — the defining property of Brownian motion.
Its power is in upgrading 'I have a continuous martingale with the right bracket' to 'I have a Brownian motion'. It is the engine behind the Dambis-Dubins-Schwarz theorem (every continuous local martingale is a time-changed Brownian motion, the time change being its own [M]), behind Girsanov's theorem (the shifted process is again Brownian because it is a continuous martingale with bracket t), and behind proving that orthogonal transformations and certain functionals of Brownian motion are again Brownian. The honest caveats that cannot be dropped: (1) Continuity is essential — the compensated Poisson process N_t - lambda t is a martingale and is NOT Brownian, but it is not continuous, and indeed its quadratic variation is N_t (a jump process), not deterministic t. (2) The bracket must be exactly t (deterministic and linear); a continuous martingale with [M]_t a different increasing process is a time-changed, not standard, Brownian motion. (3) In several dimensions the orthogonality [M^i, M^j] = 0 for i != j is what makes the components independent Brownian motions; without it you get correlated Gaussian processes.
Let B be a Brownian motion and define M_t = integral_0^t sign(B_s) dB_s. Then M is a continuous local martingale (an Ito integral) with [M]_t = integral_0^t sign(B_s)^2 ds = integral_0^t 1 ds = t. By Levy's characterization, M is itself a standard Brownian motion — even though it was manufactured from B by an unusual integral. (This M is the Brownian motion appearing in Tanaka's formula for local time.)
A continuous martingale with bracket t is automatically Brownian, however it was constructed — the recognition power of Levy's theorem.
Both hypotheses are indispensable: drop continuity and the compensated Poisson process is a counterexample; change the bracket from t to a general increasing process and you get a time-changed Brownian motion, not a standard one.