  We consider trawl processes, which are stationary and infinitely divisible
stochastic processes and can describe a wide range of statistical properties,
such as heavy tails and long memory. In this paper, we develop the first
likelihood-based methodology for the inference of real-valued trawl processes
and introduce novel deterministic and probabilistic forecasting methods. Being
non-Markovian, with a highly intractable likelihood function, trawl processes
require the use of composite likelihood functions to parsimoniously capture
their statistical properties. We formulate the composite likelihood estimation
as a stochastic optimization problem for which it is feasible to implement
iterative gradient descent methods. We derive novel gradient estimators with
variances that are reduced by several orders of magnitude. We analyze both the
theoretical properties and practical implementation details of these estimators
and release a Python library which can be used to fit a large class of trawl
processes. In a simulation study, we demonstrate that our estimators outperform
the generalized method of moments estimators in terms of both parameter
estimation error and out-of-sample forecasting error. Finally, we formalize a
stochastic chain rule for our gradient estimators. We apply the new theory to
trawl processes and provide a unified likelihood-based methodology for the
inference of both real-valued and integer-valued trawl processes.
