Multivariate stochastic delay differential equations and CAR representations of CARMA processes

TL;DR

This paper develops a new representation of multivariate CARMA processes as stochastic delay differential equations and introduces a prediction formula, supported by a general theory on existence and uniqueness of such equations.

Contribution

It introduces a novel higher-order stochastic delay differential equation representation for multivariate CARMA processes and develops a general theory for their existence, uniqueness, and representations.

Findings

Representation of CARMA as stochastic delay differential equations

Prediction formula for CARMA processes

General theory for multivariate stochastic delay differential equations

Abstract

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a continuous-time equivalent of the AR($\infty$) representation. Furthermore, we show how this representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations.