On Multiple Markov SαS Processes

Abstract

In case {X_t; t ∈ ℝ} is a Gaussian process, Mandrekar [11] and Pitt [12] have studied the structure of n—ple Markov processes. It is shown in ([11], [12]) that a Gaussian process is n—ple Markov if and only if it has the so called Goursat representation introduced in [11]. In [11], a conjecture of Lévy [9], [10] is solved (Theorem 3.12). This says that the solution of the nth order stochastic differential equation with white noise input is n—ple Markov if and only if the differential operator satisfies the so called Polya condition; namely, the Wronksians of all orders are different from zero [7] for the solutions. Recently, Adler, Cambanis, Samorodnitsky [1] have studied a representation for symmetric stable left Markov processes. Our purpose is to study n—ple Markov processes and their representations for the symmetric stable case and also to study the Markov properties of the solutions of the nth order ordinary stochastic differential equations with stationary symmetric α-stable input. The latter generalizes earlier work of Hida [6] and Lévy [10] and the former allows us to show that a process of type X_t=ϕ(t)L(τ(t)), L Lévy process and τ(t) measuring time change in [1] is first order Markov.

Supported ONR N00014–85-K-0150