On Multiple Markov SαS Processes

  • V. Mandrekar
  • B. Thelen
Part of the Progress in Probabilty book series (PRPR, volume 25)


In case {Xt; 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 Xt=ϕ(t)L(τ(t)), L Lévy process and τ(t) measuring time change in [1] is first order Markov.


Markov Process Gaussian Process Stochastic Differential Equation Gaussian Case Levy Process 
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Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • V. Mandrekar
    • 1
  • B. Thelen
    • 2
  1. 1.Department of Statistics and ProbabilityMichigan State UniversityE. LansingUSA
  2. 2.Department of StatisticsUniversity of MichiganAnn ArborUSA

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