Quadratic Fractionally Integrated Moving Average Processes with Long-Range Dependence
DOI:
https://doi.org/10.17268/sel.mat.2024.01.01Palavras-chave:
Fractionally Integrated Moving Average processes,, long-range dependence, quadratic Ornstein-Uhlenbeck type processesResumo
Stochastic processes with the long-range dependency (LRD) property are fundamental to modeling data that exhibit slow power decay of the covariance function. Such behavior often appears in the analysis of financial data, telecommunications, and various natural phenomena. Thus, introducing new stochastic models and statistical methods that take the LRD into account is of great interest. Based on previous work, we introduce a new stochastic process called quadratic fractionally integrated moving average, that arises from the Quadratic Ornstein-Uhlenbeck Type (QOUT) process, proposed in the literature. We consider Lévy noises of finite second-order moments and use a construction based on a moving average stochastic process whose kernel is that of a QOUT process. Then, using Riemann-Liouville fractional integrals, we propose a fractionally integrated moving average process, for which we highlight some results, including the LRD. We also propose the estimation of the parameters for this process for the case of fractional Brownian noise, showing its efficiency through a Monte Carlo simulation. By an application based on Brazil’s stock market prices, we illustrate how this process can be used in practice with the Sao Paulo’s Stock Exchange Index data set, also known as the BOVESPA Index.
Referências
Sato K. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press; 1999.
Marquardt T. Fractional Lévy process with an application to long memory moving average processes. Bernoulli 2006; 12(6):1099-1126.
Medino AV, Lopes SRC, Morgado R, Dorea CCY. Generalized Langevin equation driven by Lévy processes: A probabilistic, numerical and time series based approach. Physica A: Statistical Mechanics and its Applications 2012; 391(3):572-581.
Magdziarz M, Weron A. Ergodic properties of anomalous diffusion processes. Ann. of Physics 2011; 326(9):2431-2443.
Feltes GL, Lopes SRC. Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence. ALEA, Lat. Am. J. Probab. Math. Stat. 2022; 19:599-615.
Xu Y, Li Y, Zhang H, Li X, Kurths J. The switch in a genetic toggle system with Lévy noise. Scientific Reports 2016; 6:31505.
Morales M, SchoutensW. A risk model driven by Lévy processes. Applied Stochastic Models in Business and Industry 2003; 19(2):147-167.
Guo F, Wang D. Uniform asymptotic estimates for ruin probabilities of renewal risk models with exponential Lévy process investment returns and dependent claims. Applied Stochastic Models in Business and Industry 2013; 29(3):295-313.
Brockwell PJ. Recent results in the theory and applications of CARMA processes. Ann. Institute of Statistic and Mathematics 2014; 66:647-685.
Barndorff-Nielsen OE, Shephard N. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. Royal Statistic Society, Series B 2001; 63(2):167-241.
Basse A, Pedersen J. Lévy driven moving averages and semimartingales. Stochastic Processes and their Applications 2009; 119(9):2970-2991.
Belomestny D, Panov V,Woerner JHC. Low-frequency estimation of continuous-time moving average Lévy processes. Bernoulli 2019; 25(2):902-931.
Stein J, Lopes SRC, Medino AV. Continuous processes derived from the solution of generalized Langevin equation: theoretical properties and estimation. J. of Statistical Computation and Simulation 2016; 86:2819-2845.
Rajput BS, Rosinski J. Spectral representations of infinitely divisible processes. Probability Theory Related Fields 1989; 82:451-487.
Beran J, Feng Y, Ghosh S, Kulik R. Long-Memory Processes - Probabilistic Properties and Statistical Methods. Berlin: Springer-Verlag; 2013.
Lopes SRC. Long-Range Dependence in Mean and Volatility: Models, Estimation and Forecasting. In and Out of Equilibrium 2: Progress in Probability, V. Sidoravicius e M.E. Vares (Eds.), Birkhäuser; 2008; 60:497-525.
Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives. Theory and Applications. Philadelphia: Gordon and Breach; 1993.
Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. New York: Academic Press; 2007.
Leach EB. On a converse of the H¨older inequality. Proc. Americal Mathematics Society, 1956; 7:607-608.
Samorodnitsky G, Taqqu MS. Stable Non-Gaussian Random Processes. New York: Chapman & Hall; 1994.
Ivkovic Z. Deduction of Wold Representation of Stationary Processes from Cramer Representation of Second-Order Processes. The Annals of Probability, 1975; 3:546-548.
Brockwell PJ, Davis, RA. Time Series: Theory and Methods. New York: Springer; 2006.
Bender C, Lindner A, and Schicks M. Finite variation of fractional Lévy processes. J. Theory Probability, 2012; 25:594-612.
Pipiras V, Taqqu MS. Integration questions related to fractional Brownian motion. Probability Theory Related Fields 2000; 118:251-291.
Granger CWJ. The Typical Spectral Shape of an Economic Variable. Econometrica 1966; 34(1):150-161.
Ding Z, Granger CWJ, Engle R. A long memory property of stock market returns and a new model. J. of Empirical Finance 1993; 1:83–106.
Breidt FJ, Crato N, Lima PFJ. Modeling long memory stochastic volatility. Baltimore: Working Paper of John Hopkins University 1993.
Whittle P. Estimation and information in stationary time series. Arkiv Matematik, 1953; 2:423–434.
Downloads
Publicado
Como Citar
Edição
Seção
Licença
Copyright (c) 2024 Selecciones Matemáticas
Este trabalho está licenciado sob uma licença Creative Commons Attribution 4.0 International License.
Os autores que publicam nesta revista aceitam as seguintes condições:
Os autores mantêm os direitos autorais e atribuem à revista o direito da primeira publicação, com o trabalho registrado com a licença de atribuição Creative Commons Atribución 4.0 Internacional (CC BY 4.0), que permite que terceiros usem o material publicado sempre que mencionarem a autoria do trabalho e os direitos autorais. Primeira publicação nesta revista.
Os autores podem fazer outros acordos contratuais independentes e adicionais para a distribuição não exclusiva da versão do artigo publicada nesta revista (por exemplo, incluí-la em um repositório institucional ou publicá-la em um livro), desde que afirme claramente que o trabalho Foi publicado nesta revista.
É permitido e recomendado aos autores que publiquem seus trabalhos na Internet (por exemplo, em páginas institucionais ou pessoais) antes e durante o processo de revisão e publicação, pois isso pode levar a trocas produtivas e a uma disseminação maior e mais rápida do trabalho. publicado (Consultar: efeito do acesso aberto).
