Existence of \(\beta \)-weak solutions of stochastic differential equations with measurable right-hand sides (Q946096)

It is proved that there exists a \(\beta\)-weak solution of the stochastic differential equation \[ dX(t)= f(t, X(t))\,dt+ g(t, X(t))\,dW(t), \] where \(f\), \(g\) are Borel measurable locally bounded functions and \(W(t)\) is a \(d\)-dimensional Brownian motion. As a corollary it is shown that if a specified continuity condition is also satisfied, then a weak solution exists.