 Solution

For an SDE driven by Brownian motion, a strong solution of a stochastic differential equation is adapted to the completed filtration generated by a prescribed Brownian motion and satisfies the equation on that space. A weak solution of a stochastic differential equation consists of a probability space, filtration, Brownian motion, and adapted solution satisfying the equation. Uniqueness in law means that any two weak solutions with the same initial law have the same law as processes. Pathwise uniqueness means that two solutions on the same filtered space, driven by the same Brownian motion and having the same initial value, are indistinguishable.