A stochastic differential equation with a unique (up to indistinguishability) but not strong solution

Abstract

Fix a filtered probability space (Ω,F, (F _t)_≥0,P) and a Brownian motion B on that space and consider any solution process X (on Ω) to a stochastic differential equation (SDE) dX _t=f(t, X) dB_t+g(t,X) dt (1). A well-known theorem states that pathwise uniqueness implies that the solution X to SDE (1) is strong, i.e., it is adapted to the P-completed filtration generated by B. Pathwise uniqueness means that, on any filtered probability space carrying a Brownian motion and for any initial value, SDE (1) has at most one (weak) solution. We present an example that if we only assume that, for any initial value, there is at most one solution process on the given space (Ω,F, (F _t)_≥0,P), we can no longer conclude that the solution X is strong.