Noncausality and Marginalization of Markov Processes

Abstract

In this paper it is shown that a subprocess of a Markov process is markovian if a suitable condition of noncausality is satisfied. Furthermore, a markovian condition is shown to be a natural condition when analyzing the role of the horizon (finite or infinite) in the property of noncausality. We also give further conditions implying that a process is both jointly and marginally markovian only if there is both finite and infinite noncausality and that a process verifies both finite and infinite noncausality only if it is markovian. Counterexamples are also given to illustrate the cases where these further conditions are not satisfied.

A copula-based model of speculative price dynamics in discrete time

“…(15) and (16) do not imply that X is a Markov process with respect to its natural filtration. Take X i+1 = Z i + X i and Z i+1 = X i as a counterexample: (X, Z ) is a Markov process, X satisfies (15) and (16), but…”

“…Take X i+1 = Z i + X i and Z i+1 = X i as a counterexample: (X, Z ) is a Markov process, X satisfies (15) and (16), but…”

“…The importance of the concept of no-causality relies on the fact that it permits ensuring the stability of the martingale property with respect to enlarged filtrations (see [15,16]). In fact, it is a known result that if X is both an F X and an F X ,YMarkov process and it is furthermore an F X -martingale, it turns out to be an F X ,Y -martingale as well (see [4]).…”

Causality and separability

“…Florens et al (1993) (as well as Eichler, 2007 for a linear version) have actually applied this concept of separability for causality studies. Our application of this concept is more general than the previous literature in two respects:…”

“…Only the case n = 2 was considered in Florens et al (1993) and in Eichler (2007) as well. Second, we present a general framework that encompasses both the independence of σ -fields used in Florens et al (1993) and the linear correlation approach of Eichler (2007).…”