# On non-stationary solutions to MSDDEs: representations and the cointegration space

## TL;DR

This paper explores solutions to multivariate stochastic delay differential equations with stationary increments, establishing their relation to cointegration, error correction models, and defining cointegrated MCARMA processes.

## Contribution

It provides a comprehensive characterization of the cointegration space for MSDDEs and links these solutions to error correction forms and multivariate CARMA processes.

## Key findings

- MSDDE solutions with stationary increments can be expressed in error correction form.
- Complete characterization of the cointegration vectors for MSDDEs.
- Introduction of cointegrated MCARMA processes and their relation to existing models.

## Abstract

In this paper we study solutions to multivariate stochastic delay differential equations (MSDDEs) which have stationary increments, and we show that this modeling framework is in many ways similar to the discrete-time cointegrated VAR model. In particular, we observe that an MSDDE can always be written in an error correction form and, under suitable conditions, we argue that a process with stationary increments is a solution to the MSDDE if and only if it admits a certain Granger type representation. As a direct implication of these results we obtain a complete characterization of the set of cointegration vectors (the cointegration space). Finally, we exploit the relation between MSDDEs and invertible multivariate CARMA equations to define cointegrated MCARMA processes, and we discuss how this definition is related to earlier literature.