### Browsing by Author "Tudor, Ciprian"

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Item On the ARCH model with stationary liquidity(Springer Verlag, 2021-02) Voutilainen, Marko; Ilmonen, Pauliina; Torres, Soledad; Tudor, Ciprian; Viitasaari, Lauri; Department of Mathematics and Systems Analysis; Statistics and Mathematical Data Science; Universidad de Valparaíso; University of Lille; Department of Information and Service ManagementThe classical ARCH model together with its extensions have been widely applied in the modeling of financial time series. We study a variant of the ARCH model that takes account of liquidity given by a positive stationary process. We provide minimal assumptions that ensure the existence and uniqueness of the stationary solution for this model. Moreover, we give necessary and sufficient conditions for the existence of the autocovariance function. After that, we derive an AR(1) characterization for the stationary solution yielding Yule–Walker type quadratic equations for the model parameters. In order to define a proper estimation method for the model, we first show that the autocovariance estimators of the stationary solution are consistent under relatively mild assumptions. Consequently, we prove that the natural estimators arising out of the quadratic equations inherit consistency from the autocovariance estimators. Finally, we illustrate our results with several examples and a simulation study.Item Vector-valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation(WILEY-BLACKWELL, 2022-09) Voutilainen, Marko; Viitasaari, Lauri; Ilmonen, Pauliina; Torres, Soledad; Tudor, Ciprian; Department of Mathematics and Systems Analysis; Department of Information and Service Management; Statistics and Mathematical Data ScienceGeneralizations of the Ornstein-Uhlenbeck process defined through Langevin equations, such as fractional Ornstein-Uhlenbeck processes, have recently received a lot of attention. However, most of the literature focuses on the one-dimensional case with Gaussian noise. In particular, estimation of the unknown parameter is widely studied under Gaussian stationary increment noise. In this article, we consider estimation of the unknown model parameter in the multidimensional version of the Langevin equation, where the parameter is a matrix and the noise is a general, not necessarily Gaussian, vector-valued process with stationary increments. Based on algebraic Riccati equations, we construct an estimator for the parameter matrix. Moreover, we prove the consistency of the estimator and derive its limiting distribution under natural assumptions. In addition, to motivate our work, we prove that the Langevin equation characterizes essentially all multidimensional stationary processes.