Abstract. Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C * -hyperbialgebra are characterised. The characterisation is used to obtain dilations and stochastic forms of Stinespring decomposition for completely positive convolution cocycles on a C * -bialgebra.Stochastic (or Markovian) cocycles on operator algebras are basic objects of interest in quantum probability ([Acc]) and have been extensively investigated using quantum stochastic analysis (see [Lin]). There is also a well-developed theory of quantum Lévy processes, that is stationary, independent-increment, *-homomorphic processes on a *-bialgebra (see [Sch], [Fra] and references therein). Close examination of these two directions has naturally led to the notion of quantum stochastic convolution cocycle on a quantum group (or, more generally, on a coalgebra), as introduced and investigated in [ . Compact quantum hypergroups differ from compact quantum groups in that their coproduct need not be multiplicative. However, it remains completely positive, which makes compact quantum hypergroups, or more generally C * -hyperbialgebras, an appropriate category for the consideration of completely positive quantum stochastic convolution cocycles in a topological context (for the purely algebraic case see [FrS]). These cocycles may be viewed as natural counterparts of stationary, independent-increment processes on hypergroups. In [LS 4 ] it is shown that, under certain regularity conditions, they satisfy coalgebraic quantum stochastic differential equations.The aim of this paper is to prove dilation theorems for quantum stochastic convolution cocycles on a C * -bialgebra. To this end it is first necessary to establish the detailed structure of the stochastic generators of completely positive and contractive convolution cocycles. We give a direct derivation of this exploiting ideas used in the analysis of standard quantum stochastic cocycles with finite-dimensional noise space ([LiP]). Once the structure of generators is known, one may consider question of dilating completely positive convolution cocycles to * -homomorphic ones. In the context of standard quantum stochastic cocycles this problem was treated in [GLSW] and [GLW] (see also [Bel]). In the first of these papers it was shown