Noncausality in Continuous Time

In this paper, we define different concepts of noncausality for continuous-time processes,
using conditional independence and decomposition of semi-martingales. These
definitions extend the ones already given in the case of discrete-time processes. As in the
discrete-time setup, continuous-time noncausality is a property concerned with the prediction
horizon (global versus instantaneous noncausality) and the nature of the prediction
(strong versus weak noncausality). Relations between the resulting continuous-time noncausality
concepts are then studied for the class of decomposable semi-martingales, for
which, in general, the weak instantaneous noncausality does not imply the strong global
noncausality. The paper then characterizes these different concepts of noncausality in the
cases of counting processes and Markov processes.
KEYWORDS:Noncausality, continuous-time, semi-martingales, Doob-Meyer decomposition.

1. INTRODUCTION
FOLLOWINGTH E SEMINAL PAPERS by Granger (1969) and Sims (1972), the
noncausality concept plays an increasing role in Econometrics and a mostly
complete study of the relations between diverse forms of this notion has been
performed. Noncausality expressed in terms of orthogonality in the Hilbert
space of squared integrable random variables was first studied by Hosoya (1977)
and extensively treated by Florens and Mouchart (1985), while definitions in
terms of conditional independence have been given, for example, by Florens and
Mouchart (1982) and by Bouissou, Laffont, and Vuong (1986). Noncausality is,
in any case, a prediction property and the central question is: is it possible to
reduce the available information in order to predict a given stochastic process?
In these previous papers, two distinctions between various noncausality concepts
appear, sometimes implicitly. One can first oppose a one-step ahead (or instantaneous)
analysis (Granger's type definition) to a prediction property valid for
any horizon (global noncausality or Sims' type definition). On the other hand,
the definition may be focused on the prediction of the mean of the process
(weak noncausality) or of any function of the process (strong noncausality).
However, in any of these previous papers, the underlying processes are indexed
by a discrete time set, which implies, in particular, that the notion of a one-step
ahead forecast is defined unambigously.