Itô integral construction and isometry

A simple predictable integrand is constant on each interval $(t_i,t_{i+1}]$, with its coefficient measurable at time $t_i$. Its integral is the coefficient-weighted sum of the corresponding Brownian increments. Endpoints are chosen this way because the weight cannot inspect the increment it multiplies. Independence and conditional mean zero make cross terms disappear when the square of the sum is averaged. This is a probabilistic construction, not a pathwise prescription that works for arbitrary anticipative weights.

Predictability is measurability with respect to the sigma algebra generated by adapted left-continuous processes. Continuous adapted processes are predictable. A general adapted process need not automatically meet every measurability requirement for integration, so adaptedness should not replace predictability without qualification. Predictable step processes are dense in the predictable square-integrable space. Approximating there gives a controlled norm in which different step approximations converge to the same stochastic integral.

Itô isometry equates the integral's second moment with the expected time integral of the squared integrand. It first follows for step processes from the vanished cross terms. If integrands approach one another in that joint square norm, their integrals approach in mean square as well. Completeness therefore defines the integral for the limiting predictable integrand. The identity survives the limit, providing both uniqueness of the construction and a practical way to calculate variances without finding a full distribution.

The mean is zero, and applying the construction up to each intermediate time gives a martingale. A maximal inequality strengthens convergence to convergence in expected squared supremum for the resulting martingales, allowing a continuous version to be selected. This is why the terminal-value completion is consistent with a continuous process, rather than merely an unrelated random variable at each time. The integrand is identified only up to time-probability null sets, and such changes do not alter the integral.

If the squared time integral is finite almost surely but its expectation may be infinite, stop when that accumulated integral reaches a large level. The stopped integrands satisfy the square-integrable construction, and their integrals agree before the stopping times. Patching produces a continuous local martingale. Without further integrability it need not be a true martingale, so the assertion of zero expectation cannot simply be carried over. Local existence of an integral and global mean conservation are different conclusions.

A deterministic integrand has an additional Gaussian property: its integral is centered Gaussian with variance its squared time integral. A random predictable integrand generally does not produce a Gaussian random variable merely because isometry supplies its second moment. For example, integrating Brownian motion against itself will give a centered quadratic function of the terminal Brownian value. The integral remains a square-integrable martingale at each finite horizon, while its terminal distribution is visibly not a normal distribution.