  Existence and uniqueness theorems for quantum stochastic differential
equations with nontrivial initial conditions are proved for coefficients with
completely bounded columns. Applications are given for the case of
finite-dimensional initial space or, more generally, for coefficients
satisfying a finite localisability condition. Necessary and sufficient
conditions are obtained for a conjugate pair of quantum stochastic cocycles on
a finite-dimensional operator space to strongly satisfy such a quantum
stochastic differential equation. This gives an alternative approach to quantum
stochastic convolution cocycles on a coalgebra.
