= 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 and
$$
X_t=X_0+\int_0^t b(X_s)\,ds+\int_0^t\sigma(X_s)\,dW_s
$$
almost 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.
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