Strong solution of a stochastic differential equation

A strong solution is adapted to the filtration of a prescribed driving Brownian motion on a prescribed probability space and satisfies the stochastic integral equation almost surely.

A stochastic differential equation has strong existence for specified coefficients and initial data if a strong stochastic solution can be constructed on every prescribed stochastic basis carrying the required initial variable and driving Brownian motion. This existence assertion is distinct from pathwise uniqueness and from a definition of the solution itself.

For bounded Borel $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, the stochastic differential equation $d{X}_t=b(X_t)dt+d{B}_t$ has strong existence and pathwise uniqueness. The identity diffusion matrix is nondegenerate. A Girsanov theorem argument alone establishes weak existence and uniqueness in law; the strong conclusion requires an additional theorem. This statement does not extend without further hypotheses to arbitrary path-dependent drift or arbitrary diffusion matrices. See the primary bounded-drift strong-solution theorem.