  We consider the problem of modelling restricted interactions between
continuously-observed time series as given by a known static graph (or network)
structure. For this purpose, we define a parametric multivariate Graph
Ornstein-Uhlenbeck (GrOU) process driven by a general L\'evy process to study
the momentum and network effects amongst nodes, effects that quantify the
impact of a node on itself and that of its neighbours, respectively. We derive
the maximum likelihood estimators (MLEs) and their usual properties (existence,
uniqueness and efficiency) along with their asymptotic normality and
consistency. Additionally, an Adaptive Lasso approach, or a penalised
likelihood scheme, infers both the graph structure along with the GrOU
parameters concurrently and is shown to satisfy similar properties. Finally, we
show that the asymptotic theory extends to the case when stochastic volatility
modulation of the driving L\'evy process is considered.
