CONTINUOUS-TIME THRESHOLD AR(1) PROCESSES

A threshold AR(1) process with boundary width 2 delta > 0 was defined by Brockwell and Hyndman [5] in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise l inear and Lipschitz. The positive boundary-width is a convenient mathe matical device to smooth out the coefficient changes at the boundary a nd hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is der ived. In this paper we give a direct definition of a threshold AR(1) p rocess with delta = 0 in terms of the weak solution of a certain stoch astic differential equation. Two characterizations of the distribution s of the process are investigated. Both express the characteristic fun ction of the transition probability distribution as an explicit functi onal of standard Brownian motion. It is shown that the joint distribut ions of this solution with delta = 0 are the weak limits as delta down arrow 0 of the distributions of the solution with delta > 0. The sens e in which an approximating sequence of processes used by Brockwell an d Hyndman [5] converges to this weak solution is also investigated. So me numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obta ined from the Cameron-Martin-Girsanov formula and results of Engelbert and Schmidt [9]. We also derive the stationary distribution (under ap propriate assumptions) and investigate stability of these processes.