Financial Applications of Random Matrix Theory: a short review

We discuss the applications of Random Matrix Theory in the context of financial markets and econometric models, a topic about which a considerable number of papers have been devoted to in the last decade. This mini-review is intended to guide the reader through various theoretical results (the Marcenko-Pastur spectrum and its various generalisations, random SVD, free matrices, largest eigenvalue statistics, etc.) as well as some concrete applications to portfolio optimisation and out-of-sample risk estimation.Comment: To appear in the "Handbook on Random Matrix Theory", Oxford University Pres

On Some Mixing Properties of Copula-Based Markov Chains

This paper brings some insights of ψ′-mixing, ψ∗-mixing and ψ-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for ψ-mixing or ψ′-mixing. We show that perturbations of ψ′-mixing copula-based Markov chains are ψ′-mixing while perturbations of ψ-mixing Markov chains are not necessarily ψ-mixing Markov chains, even when the perturbed copula generates ψ-mixing. The Farlie–Gumbel–Morgenstern, gaussian and Ali-Mikhail-Haq copula families are considered among other examples. A statistical study is provided to emphasize the impact of perturbations on copula-based Markov chains in a simulation study. Moreover, we provide a correction to a statement made in Longla et al. (J Korean Stat Soc, 1–23, 2021) on ψ-mixing

Goodness-of-Fit tests with Dependent Observations

We revisit the Kolmogorov-Smirnov and Cram\'er-von Mises goodness-of-fit (GoF) tests and propose a generalisation to identically distributed, but dependent univariate random variables. We show that the dependence leads to a reduction of the "effective" number of independent observations. The generalised GoF tests are not distribution-free but rather depend on all the lagged bivariate copulas. These objects, that we call "self-copulas", encode all the non-linear temporal dependences. We introduce a specific, log-normal model for these self-copulas, for which a number of analytical results are derived. An application to financial time series is provided. As is well known, the dependence is to be long-ranged in this case, a finding that we confirm using self-copulas. As a consequence, the acceptance rates for GoF tests are substantially higher than if the returns were iid random variables.Comment: 26 page

Equivalence and bifurcations of finite order stochastic processes

This article presents an equivalence notion of finite order stochastic processes. Local dependence measures are defined in terms of joint and marginal densities. The dependence measures are classified topologically using level sets. The corresponding bifurcation theory is illustrated with some simple examples.