The theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, is extended to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, and automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Working in the universal enveloping von Neumann bialgebra, the stochastic generators of Markov-regular, completely positive, respectively *-homomorphic, quantum stochastic convolution cocycles are characterised. Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C*-bialgebra are characterised. Applying a recent result of Belton's, we give a thorough treatment of the approximation of quantum stochastic convolution cocycles by quantum random walks.