1,720,981 research outputs found
Multivariate estimation in nonparametric models: Stochastic neural networks and Lévy processes
Get PDFMixed orthogonality graphs for continuous‐time state space models and orthogonal projections
No full textIn this article, we derive (local) orthogonality graphs for the popular continuous‐time state space models, including in particular multivariate continuous‐time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen‐Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well‐defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well‐defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix of the driving Lévy process
On asymptotic independence in higher dimensions
Get PDFIn the study of extremes, the presence of asymptotic independence signifies
that extreme events across multiple variables are probably less likely to occur
together. Although well-understood in a bivariate context, the concept remains
relatively unexplored when addressing the nuances of joint occurrence of
extremes in higher dimensions. In this paper, we propose a notion of mutual
asymptotic independence to capture the behavior of joint extremes in dimensions
larger than two and contrast it with the classical notion of (pairwise)
asymptotic independence. Furthermore, we define -wise asymptotic
independence which lies in between pairwise and mutual asymptotic independence.
The concepts are compared using examples of Archimedean, Gaussian and
Marshall-Olkin copulas among others. Notably, for the popular Gaussian copula,
we provide explicit conditions on the correlation matrix for mutual asymptotic
independence to hold; moreover, we are able to compute exact tail orders for
various tail events.Comment: 10 pages. A short version of the discussion between pairwise and
mutual asymptotic independence mentioned in this paper appeared in a previous
version of arXiv:2309.15511 (see v1), but has been subsequently removed. The
current paper introduces this concept with additional ideas, notes and
example
Measuring risk contagion in financial networks with CoVaR
No full textThe stability of a complex financial system may be assessed by measuring risk contagion between various financial institutions with relatively high exposure. We consider a financial network model using a bipartite graph of financial institutions (e.g. banks, investment companies, insurance firms) on one side and financial assets on the other. Following empirical evidence, returns from such risky assets are modelled by heavy-tailed distributions, whereas their joint dependence is characterised by copula models exhibiting a variety of tail-dependence behaviour. We consider CoVaR, a popular measure of risk contagion, and study its asymptotic behaviour under broad model assumptions. We further propose the extreme CoVaR index (ECI) for capturing the strength of risk contagion between risk entities in such networks, which is particularly useful for models exhibiting asymptotic independence. The results are illustrated by providing precise expressions of CoVaR and ECI when the dependence of the assets is modelled using two well-known multivariate dependence structures: the Gaussian copula and the Marshall–Olkin copula
Mixed orthogonality graphs for continuous‐time state space models and orthogonal projections
Get PDFIn this article, we derive (local) orthogonality graphs for the popular continuous-time state space models, including in particular multivariate continuous-time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the
components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen-Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well-defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well-defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix
of the driving Lévy process
Measuring risk contagion in financial networks with CoVaR
Get PDFThe stability of a complex financial system may be assessed by measuring risk
contagion between various financial institutions with relatively high exposure.
We consider a financial network model using a bipartite graph of financial
institutions (e.g., banks, investment companies, insurance firms) on one side
and financial assets on the other. Following empirical evidence, returns from
such risky assets are modeled by heavy-tailed distributions, whereas their
joint dependence is characterized by copula models exhibiting a variety of tail
dependence behavior. We consider CoVaR, a popular measure of risk contagion and
study its asymptotic behavior under broad model assumptions. We further propose
the Extreme CoVaR Index (ECI) for capturing the strength of risk contagion
between risk entities in such networks, which is particularly useful for models
exhibiting asymptotic independence. The results are illustrated by providing
precise expressions of CoVaR and ECI when the dependence of the assets is
modeled using two well-known multivariate dependence structures: the Gaussian
copula and the Marshall-Olkin copula.Comment: 39 pages, 2 figures. Modifications: (1) Section 2 from previous
version with discussion on pairwise and mutual asymptotic independence is
removed (a separate note on this will appear later), (2) Title of the paper
is changed, (3) Minor modifications made to previous results and proofs, (4)
New references are adde
Partial correlation graphs for continuous-parameter time series
Get PDFIn this paper, we establish the partial correlation graph for multivariate continuous-time stochastic processes, assuming only that the underlying process is stationary and mean-square continuous with expectation zero and spectral density function. In the partial correlation graph, the vertices are the components of the process and the undirected edges represent partial correlations between the vertices. To define this graph, we therefore first introduce the partial correlation relation for continuous-time processes and provide several equivalent characterisations. In particular, we establish that the partial correlation relation defines a graphoid. The partial correlation graph additionally satisfies the usual Markov properties and the edges can be determined very easily via the inverse of the spectral density function. Throughout the paper we compare and relate the partial correlation graph to the mixed (local) orthogonality graph of Fasen-Hartmann and Schenk (Stoch Process Appl 179:104501, 2024. https://doi.org/10.1016/j.spa.2024.104501). Finally, as an example, we explicitly characterise and interpret the edges in the partial correlation graph for the popular multivariate continuous-time AR (MCAR) processes
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