h(u)B(du) and 7/ J_ iÏfe: £) implies that h(u)gi(u)du = 0 for all. J âCO. J âCO i = 1,2, .... [ 6 ] E. L. Ince, Ordinary Differential Equations, Dover, New York, 1926.
V. Mandrekar Nagoya Math. J. Vol. 54 (1974), 69-78
ON THE MULTIPLE MARKOV PROPERTY OF LEVY-HIDA FOR GAUSSIAN PROCESSES V. MANDREKAR* The purpose of this note is to clarify relations between multiple Markov properties (MMP) defined by Levy ([8], [9]) and Hida [5] for Gaussian processes and to extend some work in Levy [8] and Hida [5]. In the stationary Gaussian case it has been shown ([5], [4]) that these notions of MMP coincide. Interesting examples of (non-stationary) processes satisfying MMP can be found in [5], [8]. We now set up some notation: Let {x(t), teR} be a Gaussian stochastic process (GSP) and &t for each t be the σ-field generated by {x(τ), τ < t}. (Henceforth, . (1.6)
t)Y(t, s) - /(τ)^*3)(s, s, ί), s < τ .
Γ(τ, 8) - α(τ, ί)Γ(ί, β) = ftfB'Ks, > tί > s'N >
> si
Y(τ,s) = /(r)^*(s,s',f).
Now (1.5), (1.6) and condition (1.4) (ii) imply u(s, s', t') = u(s, s, t), τ > max(£^, tN), i.e., u(s) — u(s9 s,t) is well defined as a function s. We thus get for s < τ (1.7)
E(x(τ)\^S) = Σ fitful)
.
i =l
Clearly, for each ί, {%*(«), ^",(05), s < r} is a Martingale giving for each i,E\Ui(s)f is non-decreasing. Also using (1.7), (0.2) (i) and Jenssen inequality for conditional expectation we get for s < τ, 2f=i sup s < r E\Ui(s)\2 is finite. By Martingale convergence theorem ([3], Ch. VII) for each i, a.e. limit of Uiis) as s | τ exists and if we denote by u^τ) — lim s t r ^(s) we get {Ui(s),#Sfs < τ) is a Martingale where # r = α(Uί 0. x(t) = 2f=1 fat) Σk
