Strong and weak solutions

Given an initial variable and a Brownian motion on a filtered probability space satisfying the usual conditions, a strong solution is constructed as a nonanticipating measurable function of that initial data and the driving Brownian path. In particular, it is adapted to their completed generated filtration. We assume the initial variable is independent of future Brownian increments. The integral equation holds almost surely for every time, using continuous versions so that equality at a countable dense set extends to all times.

A process merely adapted to some larger filtration containing extra independent randomness is not automatically strong in this sense. That extra randomness may be permitted in a weak solution, but the strong existence question asks whether it is necessary. For deterministic coefficients and deterministic initial state, an explicit expression involving only past Brownian values or stochastic integrals gives a natural route to checking strong existence. It still needs to satisfy the equation and its integrability requirements, rather than merely look nonanticipating.

A weak solution consists of a probability space, a suitable filtration, a Brownian motion for that filtration, and an adapted continuous process satisfying the same integral equation. These objects are chosen together. Weak existence can therefore hold even when no solution can be constructed as a function of a specified Brownian driver alone. Changes of measure and martingale problems are useful weak-existence tools because they construct a joint probabilistic system without first supplying a pathwise solution map.

Uniqueness in law means that any two weak solutions with the same prescribed initial distribution have the same distribution as processes on continuous-path space. This is stronger than having equal one-time marginals but weaker than saying two coupled sample paths coincide. Pathwise uniqueness compares two solutions on the same space with the same Brownian driver and the same initial variable, and requires them to be indistinguishable. The shared driver is essential; two independent solutions of a well-posed equation need not be equal.

For standard finite-dimensional Itô SDEs with Borel coefficients, the Yamada–Watanabe principle says that weak existence together with pathwise uniqueness yields strong existence and uniqueness in law, under the usual solution framework. This is a theorem connecting two separately established properties, not a definition that makes them interchangeable. Discontinuous or degenerate coefficients can exhibit distinctions invisible in the globally Lipschitz setting, so conclusions should be matched to the precise hypotheses actually proved.

When comparing candidate formulas, first identify whether they use the same driver and initial value. Next check the integral identity. Finally decide whether the desired conclusion concerns a measurable construction, a coupling identity, or a law on path space. This order avoids a frequent mistake: observing that two Gaussian variables share a distribution and treating that observation as a proof that they agree almost surely. Equal laws allow many couplings, including independent and negatively correlated ones.