Filtrations, adaptedness, and predictability

A discrete filtration is an increasing sequence of sigma-algebras indexed by nonnegative integers. At time n, its events are those decidable using information available by that time. A process is adapted when its time-n variable is measurable for the time-n sigma-algebra. Integrability is an additional condition, not part of adaptedness. A martingale is adapted and integrable at each time, and its next conditional expectation equals its present value. A submartingale replaces equality with a conditional lower bound on the next value.

For independent fair signs, the natural filtration reveals the first n signs and their partial sum is a martingale. If the initial information instead reveals the first sign, that first increment has nonzero conditional value on each revealed branch. The underlying outcome probabilities have not changed, but the fairness assertion has. Adding information can invalidate a martingale property, whereas replacing a filtration by a smaller one preserves it only when the process remains adapted; the tower property then supplies the required conditional equality.

We index a weight H at time k by the increment from k minus one to k that it multiplies. Predictability means that this weight is measurable for the information at k minus one. Its product with a martingale difference therefore has conditional mean zero, provided the product is integrable. A bounded predictable weight always preserves this integrability for an integrable difference. The weight may depend on the whole past, including previous gains, and need not be deterministic or independent of earlier increments.

An adapted coefficient can already use the current sign, so adaptedness alone is too weak for a pre-increment rule. Choosing the weight equal to that same fair sign makes its product with the sign identically one. The positive mean comes from using unavailable information, not from exploiting a flaw in the conditional expectation identity. This simple check is a useful way to detect an indexing error in a stochastic sum before invoking any general theorem about transforms or integration.

A nonnegative-integer-valued random time, possibly infinite, is a stopping time when the event that it has occurred by n belongs to the information at n. The first occurrence of heads is a stopping time for coin observations. The last heads time before a fixed future horizon usually is not: deciding whether it is already past may require inspecting later tosses. A rule can be a well-defined random variable and still fail this information requirement. Its name or operational interpretation does not establish measurability.

For a stopping time T, the indicator that T is at least k is known at time k minus one, because it is the complement of the event that stopping occurred by that earlier time. This makes the stopped sum a predictable transform of the increments. By contrast, the indicator that T equals k can depend on the newly observed increment. Keeping these two indicators distinct explains why stopping a martingale at a bounded stopping time preserves fairness even though the stopping decision itself can use the current observation.