Approximate Maximum Likelihood Estimation for a Two-Threshold Lévy Process: Convergence Assessment and Data Applications

This paper presents an approach to modelling complex dynamical systems using a two-threshold Lévy process, which allows for changes in process properties when crossing critical values. The proposed model combines three key components – drift, diffusion, and jump – which together enable the description of both smooth and abrupt changes in system behaviour. An important feature is the division of the process into three regimes, which enables separate parameter estimation in each the ranges. This approach enhances the flexibility of the model and makes it suitable for analysing processes with complex hierarchical dynamics. To estimate the model parameters, an algorithm based on Approximate Maximum Likelihood Estimation (AMLE) was developed. It is implemented as an iterative procedure, which refines the parameters at each step with respect to the current threshold values and continues calculations until convergence is achieved. The algorithm enables estimation of drift parameters, diffusion coefficients and jump components, as well as the identification of the optimal thresholds. Within the study, a methodology for parameter estimation across three regimes was developed, an iterative algorithm was constructed, and its software implementation was created. The algorithm was tested on a synthetically generated dataset, which made it possible to evaluate the accuracy of parameter recovery and to examine its robustness to variations in initial conditions. Particular attention was devoted to convergence analysis: numerical experiments demonstrated that, at each step of the iterative procedure, the parameter update function reduces the differences between estimates, thereby ensuring the stability of the optimisation process. The obtained results confirm the effectiveness of the proposed estimation procedure for detecting structural changes in time series. The approach not only reproduces complex process trajectories but also identifies transition points between regimes with different dynamics. In future applications, the developed methodology may be applied to the analysis of data reflecting real economic or physical processes, including financial markets, technical systems, or natural phenomena, where threshold effects and behavioural heterogeneities are observed. Thus, the two-threshold Lévy process, combined with the iterative parameter estimation algorithm, can serve as a universal tool for studying dynamical systems with multi-component structure.