Conditional Markov chains: Properties, construction and structured dependence

Given a family (Y k , k = 1, 2, …, N) of conditional Markov chains, we construct a conditional Markov chain X = (X 1, …, X N ) such that X k , k = 1, 2, …, N, are conditional Markov chains, which are conditionally independent given the information contained in some filtration \(\mathbb{F}\), and such that for each k the conditional law of X k coincides with the conditional law of Y k . This is a new result that can be used to model different phenomena such as the gating behavior of multiple ion channels in a membrane patch, or credit ratings migrations.

A Note on Independence Copula for Conditional Markov Chains

The relatively young theory of structured dependence between stochastic processes has many real-life applications in areas including finance, insurance, seismology, neuroscience, and genetics. With this monograph, the first to be devoted to the modeling of structured dependence between random processes, the authors not only meet the demand for a solid theoretical account but also develop a stochastic processes counterpart of the classical copula theory that exists for finite-dimensional random variables. Presenting both the technical aspects and the applications of the theory, this is a valuable reference for researchers and practitioners in the field, as well as for graduate students in pure and applied mathematics programs. Numerous theoretical examples are included, alongside examples of both current and potential applications, aimed at helping those who need to model structured dependence between dynamic random phenomena.

Structured Dependence between Stochastic Processes

We define a new family of multivariate stochastic processes over a finite time horizon that we call generalised Liouville processes (GLPs). GLPs are Markov processes constructed by splitting Levy random bridges into non-overlapping subprocesses via time changes. We show that the terminal values and the increments of GLPs have generalised multivariate Liouville distributions, justifying their name. We provide various other properties of GLPs and some examples.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6959FA1773CA87D7B2FE260E1C4C3D2A/S0021900220000613a.pdf/div-class-title-generalised-liouville-processes-and-their-properties-div.pdf

Generalised liouville processes and their properties

Martingale Problem: Some New Results Needed in this Book

In this paper we connect Archimedean survival processes (ASPs) with the theory of Markov copulas. ASPs were introduced by Hoyle and Menguturk (2013) to model the realized variance of two assets. We present some new properties of ASPs related to their dependency structure. We study weak and strong Markovian consistency properties of ASPs. An ASP is weak Markovian consistent, but generally not strong Markovian consistent. Our results contain necessary and sufficient conditions for an ASP to be strong Markovian consistent. These properties are closely related to the concept of Markov copulas, which is very useful in modelling different dependence phenomena. At the end we present possible applications.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0021900216000085

Adam Pytel

Papers

The Markov consistency of Archimedean survival processes