Statistical Topics and Stochastic Models for Dependent Data with Applications

6 Time Changes and Stationarity Issues for Continuous Time Autoregressive Processes of Order p

Smooth time transforms of scalar continuous-time autoregressive (AR) processes of order one, say CAR₁(p), with time-dependent coefficients are shown here to be CAR₁(p) processes also, with explicit underlying stochastic differential equations (SDE). Necessary and sufficient conditions are given for a CAR₁(p) process to be stationary or to become stationary through a pertinent time change. For that purpose, a new criterion for stationarity is given, which is proven to be equivalent with the classical notion of asymptotic stability. Many results are obtained thanks to a thorough study of the associated multivariate CAR_p(1) process. Several examples are presented; in particular, multiplicative stationarity – involving logarithmic time change – is detailed. Illustration through simulation is provided.

6.1. Introduction

Transforming a stochastic process into another one through a deformation of the index set may lead to a full class of new properties for the resulting process. This concept was formalized in Perrin and Senoussi (1999), after having been introduced in geostatistics – statistics related to continuous-time random fields – in an application to environmental data in Sampson and Guttorp (1992). For time processes, classical time changes are considered in Gray and Zhang (1988), with a particular interest to the logarithmic time change, leading to the so-called notion of multiplicative ...