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We consider trawl processes, which are stationary and infinitely divisiblestochastic processes and can describe a wide range of statistical properties,such as heavy tails and long memory. In this paper, we develop the firstlikelihood-based methodology for the inference of real-valued trawl processesand introduce novel deterministic and probabilistic forecasting methods. Beingnon-Markovian, with a highly intractable likelihood function, trawl processesrequire the use of composite likelihood functions to parsimoniously capturetheir statistical properties. We formulate the composite likelihood estimationas a stochastic optimization problem for which it is feasible to implementiterative gradient descent methods. We derive novel gradient estimators withvariances that are reduced by several orders of magnitude. We analyze both thetheoretical properties and practical implementation details of these estimatorsand release a Python library which can be used to fit a large class of trawlprocesses. In a simulation study, we demonstrate that our estimators outperformthe generalized method of moments estimators in terms of both parameterestimation error and out-of-sample forecasting error. Finally, we formalize astochastic chain rule for our gradient estimators. We apply the new theory totrawl processes and provide a unified likelihood-based methodology for theinference of both real-valued and integer-valued trawl processes.

Likelihood-based inference and forecasting for trawl processes: a stochastic optimization approach