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.

翻译：我们考虑拖网过程，这是一种平稳且无限可分的随机过程，能够描述重尾分布和长记忆性等广泛统计特性。本文首次发展了基于似然方法的实值拖网过程推断方法，并提出了新颖的确定性与概率性预测方法。由于拖网过程具有非马尔可夫性且似然函数高度复杂，需使用复合似然函数以简洁地捕捉其统计特性。我们将复合似然估计问题转化为随机优化问题，使其能够通过迭代梯度下降法实现。我们推导了新颖的梯度估计量，其方差可降低数个数量级。分析了这些估计量的理论性质与实现细节，并发布了适用于拟合一大类拖网过程的Python库。仿真研究表明，在参数估计误差与样本外预测误差方面，我们的估计量均优于广义矩估计方法。最后，我们形式化了梯度估计量的随机链式法则，并将新理论应用于拖网过程，为实值及整数值拖网过程的推断提供了统一的基于似然的方法体系。