Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators

Dalessandro, A. & Peters, G.W. Methodology and Computing in Applied Probability (2017). doi:10.1007/s11009-017-9545-8

35 Pages Posted: 26 Feb 2015 Last revised: 4 Sep 2019

See all articles by Antonio Dalessandro

Antonio Dalessandro

University College London

Gareth Peters

Department of Actuarial Mathematics and Statistics, Heriot-Watt University; University College London - Department of Statistical Science; University of Oxford - Oxford-Man Institute of Quantitative Finance; London School of Economics & Political Science (LSE) - Systemic Risk Centre; University of New South Wales (UNSW) - Faculty of Science

Date Written: January 2017

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.

Keywords: Martingale Representation; Semimartingales Decomposition; Copula Infinitesimal Generators

JEL Classification: C00, C14, C60, C63

Suggested Citation

Dalessandro, Antonio and Peters, Gareth, Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators (January 2017). Dalessandro, A. & Peters, G.W. Methodology and Computing in Applied Probability (2017). doi:10.1007/s11009-017-9545-8, Available at SSRN: https://ssrn.com/abstract=2569134 or http://dx.doi.org/10.2139/ssrn.2569134

Antonio Dalessandro (Contact Author)

University College London ( email )

Gower Street
London, WC1E 6BT
United Kingdom

Gareth Peters

Department of Actuarial Mathematics and Statistics, Heriot-Watt University ( email )

Edinburgh Campus
Edinburgh, EH14 4AS
United Kingdom

HOME PAGE: http://garethpeters78.wixsite.com/garethwpeters

University College London - Department of Statistical Science ( email )

1-19 Torrington Place
London, WC1 7HB
United Kingdom

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

University of Oxford Eagle House
Walton Well Road
Oxford, OX2 6ED
United Kingdom

London School of Economics & Political Science (LSE) - Systemic Risk Centre ( email )

Houghton St
London
United Kingdom

University of New South Wales (UNSW) - Faculty of Science ( email )

Australia

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