 Solution

Fix the initial state $x$, or a prescribed initial distribution. A weak stochastic solution consists of a filtered probability space, a Brownian motion $W$ in that filtration, and a continuous adapted process $X$, with the prescribed initial law, such that the integrals exist andalmost surely for every $t$. The integrability conditions on each finite interval are ${\int }_0^t|b(X_s)|ds<\infty$ and ${\int }_0^t\sigma (X_s{)}^{2}ds<\infty$ almost surely. The probability space and driving Brownian motion are part of what may be chosen.

A strong stochastic solution is constructed on a space carrying a specified driving Brownian motion and specified initial variable. It satisfies the same equation and is adapted to the completed filtration generated by that initial variable and the Brownian motion. Equivalently, it is a nonanticipating measurable function of those data, requiring no additional randomness. For a random initial variable, it is independent of future Brownian increments, as required by the Brownian property of the enlarged filtration.