A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same L´evy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p-1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.Peer ReviewedPostprint (published version

A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same L´evy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p-1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.Peer ReviewedPostprint (published version

Embedding in law of discrete time ARMA processes in continuous time stationary processes

© 2018. This version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Given any stationary time series {Xn : n ¿ Z} satisfying an ARMA(p, q) model for arbitrary p and q with infinitely divisible innovations, we construct a continuous time stationary process {xt : t ¿ R} such that the distribution of {xn : n ¿ Z}, the process sampled at discrete time, coincides with the distribution of {Xn}. In particular the autocovariance function of {xt } interpolates that of {Xn}.Peer ReviewedPostprint (author's final draft

Towards a sharp estimation of transfer entropy for identifying causality in financial time series

We present an improvement of an estimator of causality in financial time series via transfer entropy, which includes the side information that may affect the cause-effect relation in the system, i.e. a conditional information-transfer based causality. We show that for weakly stationary time series the conditional transfer entropy measure is nonnegative and bounded below by the Geweke's measure of Granger causality. We use k-nearest neighbor distances to estimate entropy and approximate the distribution of the estimator with bootstrap techniques. We give examples of the application of the estimator in detecting causal effects in a simulated autoregressive stationary system in three random variables with linear and non-linear couplings; in a system of non stationary variables; and with real financial data.Postprint (published version

American and exotic options in a market with frictions

In a market with frictions, bid and ask prices are described by sublinear pricing functionals, which can be defined recursively using coherent risk measures. We prove the convergence of bid and ask prices for various European and American possible path-dependent options, in particular plain vanilla, Asian, lookback and barrier options in a binomial model with transaction costs. We perform several numerical experiments to confirm the theoretical findings. We apply the results to real market data of American options and compute an implied liquidity to describe the bidask spread. This method describes liquidity over time very well, compared to the classical approach of describing bid and ask prices by quoting bid and ask implied volatilities.Peer ReviewedPostprint (author's final draft

Estimated Covid-19 burden in Spain: ARCH underreported non-stationary time series

Background The problem of dealing with misreported data is very common in a wide range of contexts for different reasons. The current situation caused by the Covid-19 worldwide pandemic is a clear example, where the data provided by official sources were not always reliable due to data collection issues and to the high proportion of asymptomatic cases. In this work, a flexible framework is proposed, with the objective of quantifying the severity of misreporting in a time series and reconstructing the most likely evolution of the process. Methods The performance of Bayesian Synthetic Likelihood to estimate the parameters of a model based on AutoRegressive Conditional Heteroskedastic time series capable of dealing with misreported information and to reconstruct the most likely evolution of the phenomenon is assessed through a comprehensive simulation study and illustrated by reconstructing the weekly Covid-19 incidence in each Spanish Autonomous Community. Results Only around 51% of the Covid-19 cases in the period 2020/02/23–2022/02/27 were reported in Spain, showing relevant differences in the severity of underreporting across the regions. Conclusions The proposed methodology provides public health decision-makers with a valuable tool in order to improve the assessment of a disease evolution under different scenarios.Research funded by Fundación MAPFRE. This work was partially supported by grant RTI2018-096072-B-I00 from the Spanish Ministry of Science and Innovation and by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020–001084-M). A.F-F acknowledges Agencia Estatal de Investigación for the financial support IJC2020-045188I/AEI/10.13039/501100011033. AC was partially financed by PID2021-123733NB-I00 (Ministerio de Ciencia e Innovación, Spain). AC and AA were partially supported by Project “EcoDep” CY-AAP2020-0000000013 (“Investissements d’Avenir” ANR-16-IDEX-0008, France).Peer ReviewedPostprint (published version

Cumulated burden of Covid-19 in Spain from a Bayesian perspective

Background The main goal of this work is to estimate the actual number of cases of Covid-19 in Spain in the period 01-31-2020/06-01-2020 by Autonomous Communities. Based on these estimates, this work allows us to accurately re-estimate the lethality of the disease in Spain, taking into account unreported cases. Methods A hierarchical Bayesian model recently proposed in the literature has been adapted to model the actual number of Covid-19 cases in Spain. Results The results of this work show that the real load of Covid-19 in Spain in the period considered is well above the data registered by the public health system. Specifically, the model estimates show that, cumulatively until June 1st, 2020, there were 2 425 930 cases of Covid-19 in Spain with characteristics similar to those reported (95% credibility interval: 2 148 261 2 813 864), from which were actually registered only 518 664. Conclusions Considering the results obtained from the second wave of the Spanish seroprevalence study, which estimates 2 350 324 cases of Covid-19 produced in Spain, in the period of time considered, it can be seen that the estimates provided by the model are quite good. This work clearly shows the key importance of having good quality data to optimize decision-making in the critical context of dealing with a pandemic.Peer ReviewedPostprint (author's final draft

Embedding in law of discrete time ARMA processes in continuous time stationary processes

© 2018. This version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Given any stationary time series {Xn : n ¿ Z} satisfying an ARMA(p, q) model for arbitrary p and q with infinitely divisible innovations, we construct a continuous time stationary process {xt : t ¿ R} such that the distribution of {xn : n ¿ Z}, the process sampled at discrete time, coincides with the distribution of {Xn}. In particular the autocovariance function of {xt } interpolates that of {Xn}.Peer Reviewe

An alternative to CARMA models via iterations of Ornstein–Uhlenbeck processes

We present a new construction of continuous ARMA processes based on iterating an Ornstein–Uhlenbeck operator OUκ that maps a random variable y(t) onto OUκy(t)=∫t−∞e−κ(t−s)dy(s). This construction resembles the procedure to build an AR( p) from an AR(1) and derives in a parsimonious model for continuous autoregression, with fewer parameters to compute than the known CARMA obtained as a solution of a system of stochastic differential equations. We show properties of this operator, give state space representation of the iterated Ornstein–Uhlenbeck process and show how to estimate the parameters of the model.Postprint (published version

A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same L´evy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p-1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.Peer Reviewe