IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v20y2018i1d10.1007_s11009-017-9545-8.html
   My bibliography  Save this article

Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators

Author

Listed:
  • Antonio Dalessandro

    (University College London)

  • Gareth W. Peters

    (University College London)

Abstract

In this paper we develop a class of applied probabilistic continuous time but discretized state space decompositions of the characterization of a multivariate generalized diffusion process. This decomposition is novel and, in particular, it allows one to construct families of mimicking classes of processes for such continuous state and continuous time diffusions in the form of a discrete state space but continuous time Markov chain representation. Furthermore, we present this novel decomposition and study its discretization properties from several perspectives. This class of decomposition both brings insight into understanding locally in the state space the induced dependence structures from the generalized diffusion process as well as admitting computationally efficient representations in order to evaluate functionals of generalized multivariate diffusion processes, which is based on a simple rank one tensor approximation of the exact representation. In particular, we investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale approach. We show that it is possible to represent, in both continuous and discrete space, that a multidimensional correlated generalized diffusion is a linear combination of processes originated from the decomposition of the starting multidimensional semimartingale. This result not only reconciles with the existing theory of diffusion approximations and decompositions, but defines the general representation of infinitesimal generators for both multidimensional generalized diffusions and, as we will demonstrate, also for the specification of copula density dependence structures. This new result provides immediate representation of the approximate weak solution for correlated stochastic differential equations. Finally, we demonstrate desirable convergence results for the proposed multidimensional semimartingales decomposition approximations.

Suggested Citation

  • Antonio Dalessandro & Gareth W. Peters, 2018. "Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 237-271, March.
    • Handle: RePEc:spr:metcap:v:20:y:2018:i:1:d:10.1007_s11009-017-9545-8
    DOI: 10.1007/s11009-017-9545-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-017-9545-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-017-9545-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Marco Scarsini, 1984. "Strong measures of concordance and convergence in probability," Post-Print hal-00542387, HAL.
    2. Marco Scarsini, 1984. "On measures of concordance," Post-Print hal-00542380, HAL.
    3. Huamin Zhang & Feng Ding, 2013. "On the Kronecker Products and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-8, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Plischke, Elmar & Borgonovo, Emanuele, 2019. "Copula theory and probabilistic sensitivity analysis: Is there a connection?," European Journal of Operational Research, Elsevier, vol. 277(3), pages 1046-1059.
    2. Edoardo Berton & Lorenzo Mercuri, 2021. "An Efficient Unified Approach for Spread Option Pricing in a Copula Market Model," Papers 2112.11968, arXiv.org, revised Feb 2023.
    3. Jiří Dvořák & Tomáš Mrkvička, 2022. "Graphical tests of independence for general distributions," Computational Statistics, Springer, vol. 37(2), pages 671-699, April.
    4. Liebscher Eckhard, 2017. "Copula-Based Dependence Measures For Piecewise Monotonicity," Dependence Modeling, De Gruyter, vol. 5(1), pages 198-220, August.
    5. Dalia Valencia & Rosa E. Lillo & Juan Romo, 2019. "A Kendall correlation coefficient between functional data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(4), pages 1083-1103, December.
    6. Sergio Ocampo, 2019. "A task-based theory of occupations with multidimensional heterogeneity," 2019 Meeting Papers 477, Society for Economic Dynamics.
    7. Hofert, Marius & Oldford, Wayne, 2018. "Visualizing dependence in high-dimensional data: An application to S&P 500 constituent data," Econometrics and Statistics, Elsevier, vol. 8(C), pages 161-183.
    8. Liebscher, Eckhard, 2021. "Kendall regression coefficient," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    9. Ferreira Helena & Ferreira Marta, 2020. "Multivariate medial correlation with applications," Dependence Modeling, De Gruyter, vol. 8(1), pages 361-372, January.
    10. Jia-Han Shih & Takeshi Emura, 2019. "Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula," Statistical Papers, Springer, vol. 60(4), pages 1101-1118, August.
    11. Manuela Schreyer & Roland Paulin & Wolfgang Trutschnig, 2017. "On the exact region determined by Kendall's τ and Spearman's ρ," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 613-633, March.
    12. Ferreira Helena & Ferreira Marta, 2020. "Multivariate medial correlation with applications," Dependence Modeling, De Gruyter, vol. 8(1), pages 361-372, January.
    13. Vanderford Courtney & Sang Yongli & Dang Xin, 2020. "Two symmetric and computationally efficient Gini correlations," Dependence Modeling, De Gruyter, vol. 8(1), pages 373-395, January.
    14. Silvia Terzi & Luca Moroni, 2022. "Local Concordance and Some Applications," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 161(2), pages 457-470, June.
    15. Kamil Gala, 2015. "On the probability distribution of the present value of benefits in multiple life insurance," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 37, pages 13-38.
    16. Machová Renáta & Korcsmáros Enikő & Marča Roland & Esseová Monika, 2022. "An International Analysis of Consumers’ Consciousness During the Covid-19 Pandemic in Slovakia and Hungary," Folia Oeconomica Stetinensia, Sciendo, vol. 22(1), pages 130-151, June.
    17. Gijbels, Irène & Kika, Vojtěch & Omelka, Marek, 2021. "On the specification of multivariate association measures and their behaviour with increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    18. Fuchs Sebastian, 2016. "A Biconvex Form for Copulas," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-13, February.
    19. Yanqin Fan & Marc Henry, 2020. "Vector copulas," Papers 2009.06558, arXiv.org, revised Apr 2021.
    20. Neslehová, Johanna, 2007. "On rank correlation measures for non-continuous random variables," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 544-567, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metcap:v:20:y:2018:i:1:d:10.1007_s11009-017-9545-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.