= Solution
A \b[<weak solution of a stochastic differential equation>] consists of a <filtered probability space>, an <adapted> continuous process $X$, and a <Brownian motion> $B$ relative to that <filtration>, with specified initial distribution, such that for every finite $t$ the coefficient integrals are finite and
$$
\boxed{X_t=X_0+\int_0^t\sigma(X_s)\,dB_s+\int_0^t b(X_s)\,ds\quad\text{almost surely for all }t.}
$$
In one dimension the integrability conditions are $\int_0^t\sigma(X_s)^2ds<\infty$ and $\int_0^t|b(X_s)|ds<\infty$. The noise integral is the <Itô integral>. A solution on a prescribed stochastic basis with prescribed $B$ is distinguished from this weak construction.
A \b[<strong solution of a stochastic differential equation>] uses the prescribed $B$ and initial variable, and is <adapted> to their augmented <natural filtration>; thus no additional randomness is needed. For deterministic $X_0=x$, each $X_t$ is determined by the driving <Brownian motion> through time $t$. Merely being <adapted> to a larger <filtration> is insufficient for this definition.
\b[<Uniqueness in law>] means that any two <weak stochastic solutions> with the same initial distribution have the same law of $X$ on continuous path space. \b[<Pathwise uniqueness>] means that two solutions on one stochastic basis, with the same $B$ and almost surely equal initial values, are <indistinguishable stochastic processes>. The latter compares paths under the same noise, rather than just comparing their <probability distributions>.
Back to article page
