a weak solution

A weak solution of dX = b(X) dt + sigma(X) dB relaxes the demand of a strong solution: instead of using a Brownian motion you are handed, you are allowed to construct the probability space, the filtration, the Brownian motion B, and the process X all together, so that the pair (X, B) satisfies the SDE. You are not required to make X a function of a pre-given B. The question it answers is: when there is no pathwise-defined solution, can we still build a process with the right diffusive dynamics? Often the answer is yes, and the weak solution is all you need, because many questions only depend on the law of X.

Formally a weak solution is a quadruple (Omega, F, P, (F_t)) together with adapted processes (X_t, B_t) on it, where B is an (F_t)-Brownian motion and X satisfies the integral equation a.s. The key relaxation is that the same SDE can have weak solutions on different spaces, and what is fixed is only the law of X (its finite-dimensional distributions / its law on path space). Weak existence can be proved under far milder assumptions than strong existence — for instance, by the martingale-problem method (Stroock-Varadhan): a continuous process X with X_0 = x is a weak solution iff f(X_t) - f(X_0) - integral_0^t L f(X_s) ds is a martingale for all smooth test functions f, where L is the diffusion generator. One can also build weak solutions by Girsanov's theorem (change the drift by a change of measure) or by Skorokhod's existence theorem (a tightness/limit argument requiring only continuity and boundedness of the coefficients).

Weak solutions are the right object whenever you only care about distributions: computing E[g(X_T)], pricing a derivative, characterizing the diffusion as a Markov process. The associated notion of uniqueness is uniqueness in law (any two weak solutions have the same law), which is weaker than pathwise uniqueness. The honest point — made precise by Yamada-Watanabe — is that weak existence plus pathwise uniqueness together force the existence of a strong solution; but weak existence alone does not, and there are SDEs with weak solutions and uniqueness in law but no strong solution at all (Tanaka's equation again).