Square-integrable martingales carry a Hilbert-space geometry. Their increments are orthogonal in \(L^2\); realized squared increments form quadratic variation; and conditional squared increments form its predictable compensator. These objects separate observed path variation from its conditional variance budget.
Orthogonal martingale increments
Proposition. Let \(M_n\in L^2\) be a martingale and \(\Delta M_k=M_k-M_{k-1}\). If \(j<k\), then \[ \mathbb E[\Delta M_j\Delta M_k]=0. \] More generally, \(\Delta M_k\) is orthogonal to every \(Y\in L^2(\mathcal F_{k-1})\).
Proof. Since \(\mathbb E[\Delta M_k\mid\mathcal F_{k-1}]=0\), \[ \mathbb E[Y\Delta M_k] =\mathbb E\!\left[ Y\,\mathbb E(\Delta M_k\mid\mathcal F_{k-1})\right]=0. \] Take \(Y=\Delta M_j\), which is \(\mathcal F_j\subseteq\mathcal F_{k-1}\)-measurable.
Theorem (martingale Pythagoras identity). For \(m<n\), \[ \mathbb E|M_n-M_m|^2 =\sum_{k=m+1}^n\mathbb E(\Delta M_k)^2. \] In particular, \[ \mathbb EM_n^2 =\mathbb EM_0^2+\sum_{k=1}^n\mathbb E(\Delta M_k)^2. \]
Proof. Expand \(M_n-M_m=\sum_{k=m+1}^n\Delta M_k\). All cross terms vanish by orthogonality.
Corollary. A square-integrable martingale converges in \(L^2\) if and only if \[ \sum_{k=1}^{\infty}\mathbb E(\Delta M_k)^2<\infty, \] equivalently \(\sup_n\mathbb EM_n^2<\infty\).
Optional and predictable quadratic variation
Definition. For \(M_0=0\), define \[ [M]_n=\sum_{k=1}^n(\Delta M_k)^2, \qquad \langle M\rangle_n =\sum_{k=1}^n \mathbb E[(\Delta M_k)^2\mid\mathcal F_{k-1}]. \] The first quantity is the realized, or optional, quadratic variation. The second is predictable quadratic variation, also called the bracket.
Theorem. Both \[ M_n^2-[M]_n \quad\text{and}\quad M_n^2-\langle M\rangle_n \] are martingales. Consequently, \([M]_n-\langle M\rangle_n\) is a martingale and \[ \mathbb E[M]_n=\mathbb E\langle M\rangle_n=\mathbb EM_n^2. \]
Proof. Since \[ \Delta(M_n^2) =2M_{n-1}\Delta M_n+(\Delta M_n)^2, \] subtracting \(\Delta[M]_n=(\Delta M_n)^2\) leaves \(2M_{n-1}\Delta M_n\), whose conditional mean is zero. Subtracting instead \[ \Delta\langle M\rangle_n =\mathbb E[(\Delta M_n)^2\mid\mathcal F_{n-1}] \] also leaves a conditionally centered increment. The expectation identities follow at time \(n\).
The distinction matters when conditional variances are random. The path records \([M]_n\); before observing increment \(n\), the process forecasts its variance through \(\Delta\langle M\rangle_n\). Martingale central limit theorems normalize by this predictable variance.
Covariation and orthogonality
Definition. For square-integrable martingales \(M,N\) starting at zero, define \[ [M,N]_n=\sum_{k=1}^n\Delta M_k\Delta N_k, \] \[ \langle M,N\rangle_n =\sum_{k=1}^n \mathbb E[\Delta M_k\Delta N_k\mid\mathcal F_{k-1}]. \] Polarization gives \[ [M,N]=\frac14([M+N]-[M-N]), \] and similarly for predictable brackets.
Proposition. The process \(M_nN_n-\langle M,N\rangle_n\) is a martingale. If \(\langle M,N\rangle_n=0\) for every \(n\), the martingales are strongly orthogonal.
Proof. Use \[ \Delta(MN)_n =M_{n-1}\Delta N_n+N_{n-1}\Delta M_n +\Delta M_n\Delta N_n. \] The first two terms have conditional mean zero; the conditional mean of the last is \(\Delta\langle M,N\rangle_n\).
Isometry for martingale transforms
Theorem. Let \(H_k\) be predictable and suppose \(\sum_{k\le n}\mathbb E[H_k^2(\Delta M_k)^2]<\infty\). Then \[ \mathbb E(H\mathbin{\cdot}M)_n^2 = \mathbb E\sum_{k=1}^nH_k^2(\Delta M_k)^2 = \mathbb E\sum_{k=1}^nH_k^2 \Delta\langle M\rangle_k. \]
Proof. The transformed increments \(H_k\Delta M_k\) are orthogonal, so Pythagoras gives the first equality. For the second, \[ \mathbb E[H_k^2(\Delta M_k)^2] = \mathbb E\!\left[ H_k^2\mathbb E((\Delta M_k)^2\mid\mathcal F_{k-1})\right]. \]
Example (simple random walk). For \(M_n=\sum_{k=1}^n\xi_k\) with independent symmetric \(\xi_k\in\{-1,1\}\), \[ [M]_n=\langle M\rangle_n=n. \] For conditionally heteroskedastic differences \(\Delta M_k=\sigma_k\varepsilon_k\), where \(\sigma_k\in\mathcal F_{k-1}\) and \(\mathbb E[\varepsilon_k^2\mid\mathcal F_{k-1}]=1\), \[ [M]_n=\sum_k\sigma_k^2\varepsilon_k^2, \qquad \langle M\rangle_n=\sum_k\sigma_k^2. \] The first is realized variation; the second is its predictable conditional mean.