  Strong solutions of p-dimensional stochastic differential equations that can
be represented locally in explicit simulation form are considered. The
following three-way equivalence is established: 1) There exists such a
representation from all starting points, 2) the representation pair satisfies a
set differential equations, and 3) the stochastic differential equation
coefficients satisfy commutation relations. Next, construction theorems, based
on a diffeomorphism between the original equation solutions and the strong
solutions to a simpler Ito integral equation, with a possible deterministic
component, are given. Finally, motivating examples are provided and reference
to its importance in filtering and option pricing is given.
