Is there an "initial condition" for ARMA model?

Question

1. ARMA model is a stochastic version of recursive relation. For deterministic recursive relations, we solve them and need initial conditions to fully get the solution. So I wonder what is an "initial condition" for ARMA model like?
2. In Introduction to Time Series and Forecasting, by Peter J. Brockwell, Richard A. Davis, they define an ARMA model to be causal, if its output process can be represented as MA($\infty$) (see equation (3.1.5)). In equation (3.1.7), they gave the explicit formulation of the MA($\infty$) representation. I don't find they specify or use any "initial" condition in the definition of causality and in the derivation of the MA($\infty$) representation. This is quite different from that we need initial conditions to fully get the solution to a deterministic recursive relation. So how shall we understand the MA($\infty$) representation of an ARMA output process?

Thanks!

Answer 1

The ARMA model in $MA(\infty)$ representation it is not a full solution. This means that based on $\phi(z)\psi(z)=\theta(z)$ you can only derive the the coefficients of $\psi(z)$, obviously when the ARMA model is causal. A full solution means to find an explicit computable formula, which can't be obtained if you don't assume initial conditions, as you already stated.