= Solution
Specify an initial value or initial <probability distribution>. A <strong solution of a stochastic differential equation> uses a prescribed <Brownian motion> on a prescribed <filtered probability space>. It is continuous, adapted to the completed <filtration> generated by that <Brownian motion> and the initial value, and satisfies
$$
X_t=X_0+\int_0^tb(X_s)\,ds+\int_0^t\sigma(X_s)\,dB_s
$$
<almost surely> for every $t$, with $\int_0^t(|b(X_s)|+\sigma(X_s)^2)ds<\infty$ <almost surely>. The given initial variable is independent of future <Brownian motion> increments. Some courses allow a larger prescribed <filtration> in their definition of a <strong solution of a stochastic differential equation>; the Brownian-generated convention states explicitly what absence of extra randomness means.
A <weak solution of a stochastic differential equation> consists of a <filtered probability space>, a <Brownian motion> relative to its <filtration>, and a continuous <adapted process> $X$ satisfying the same integrability and integral equation. Here the space and noise are part of the unknown; the <filtration> may contain randomness beyond the initial variable and the driving path.
<Pathwise uniqueness> means that on any common <filtered probability space>, any two solutions with the same driving <Brownian motion> and the same initial value are indistinguishable. <Uniqueness in law> means that all <weak solutions> with the same initial <probability distribution> have the same law as path-valued <random variables>, even on different spaces. \b[Pathwise uniqueness compares shared-noise paths; uniqueness in law compares path distributions.] Neither uniqueness notion by itself asserts existence.
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