Discrete stochastic integrals and isometries

Let M be a real square-integrable discrete martingale, starting at zero, and let H at step k be measurable for the information at k minus one. Define its integral as the cumulative sum of H times the corresponding martingale increments, starting at zero. Each product is measurable at the current time. If H is bounded on every finite horizon, conditional centering and integrability follow immediately. Unbounded coefficients are also allowed, but require a condition that controls the actual weighted increments rather than just the coefficients in isolation.

For a fixed finite horizon, assume the expected sum of squared coefficients times predictable bracket increments is finite. Taking conditional expectations shows this equals the sum of expected squares of the weighted martingale increments. Each product is therefore square integrable and has conditional mean zero. The resulting integral process is an L2 martingale. Merely saying that H is adapted or measurable eventually does not suffice: a coefficient that sees its own upcoming increment can alter its conditional mean.

Expand the square of the finite integral. Cross terms between different steps have expectation zero because the later weighted increment has zero conditional mean and the earlier one is already measurable. The remaining diagonal terms are expectations of squared weighted increments. Since the coefficient is known before the step, conditioning moves its square outside and replaces the increment square by the predictable bracket increment. This proves the discrete isometry exactly, with no approximation or independence assumption beyond the martingale conditions.

The same calculation at each step identifies the predictable bracket of the integral as the sum of the squared coefficients against the predictable bracket of M. Its realized square bracket instead sums the squared coefficients against the realized square bracket. These are pathwise finite-sum identities under the stated conventions. They show why a coefficient's ordinary unweighted L2 norm is not always the appropriate energy: the driving martingale may have history-dependent conditional variance, so some times or branches carry more variation than others.

If the expected total energy summed over all steps is finite, the integral partial sums are L2 bounded and hence converge almost surely and in L2. The isometry then identifies the terminal second moment and the mean-square tails. This is a useful completion procedure: approximating predictable integrands in the bracket-weighted energy norm makes their integral terminal values converge in L2. It does not claim convergence from a bound on individual coefficients when infinitely many increments continue to contribute energy.

A stopped martingale is a special integral with the predictable indicator that stopping has not happened before the current step. This connects optional sampling with stochastic integration. Another special case uses the previous martingale value as coefficient and yields half the difference between the current square and realized bracket, by expanding squares. That coefficient is predictable but can be unbounded; applying an L2 isometry to it still requires the corresponding higher-energy integrability. An algebraically valid pathwise sum and an L2 theorem should not be conflated.