Suppose the process
taking values in
is given by the
SDE
where the
is a matrix valued adapted stochastic process,
is a column-valued adapted process and the
is the column of the standard Brownian motions. The process
taking values in
and given by the
SDE
has the same component-wise distributions as
if the deterministic functions
and
are given by the
relationships
The
is a
-th
row of the matrix
and the multiplication
above is the scalar product.