Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators

TL;DR

This paper presents a novel tensor-based approach to approximate and decompose multidimensional correlated diffusions, providing new insights into dependence structures and enabling better solutions for stochastic differential equations.

Contribution

It introduces a new tensor approximation method for generalized correlated diffusions and offers a unified representation of their infinitesimal generators and copula dependence structures.

Findings

Proposes a tensor approximation framework for multidimensional diffusions.

Provides convergence results for the semimartingale decomposition approximations.

Demonstrates the representation of copula dependence structures in diffusions.

Abstract

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 that originate 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 solution for correlated stochastic differential equations. We demonstrate desirable convergence results for the proposed multidimensional semimartingales decomposition approximations.