second-order Gauss-Markov process

Let X(t) denote a continuous-time random process. If X(t) is stationary,
Gaussian, Markovian and continuous in probability, then X(t) satisfies
the following linear stochastic differential equation [1,2,3]:

dX(t) = -rho*(X_t - mu)*dt + sigma*dW(t)

where mu, sigma & rho are constants and W(t) is Brownian motion.
X(t) is often called an Ornstein-Uhlenbeck process or a
first-order Gauss-Markov process.

A process Y(t) is second-order Markovian if, for every k and
every set of timepoints t_1 < t_2 < ... < t_k, it is true that

P( Y(t_k) < y | Y(t_{k-1}), Y(t_{k-2}), ..., Y(t_2), Y(t_1) ) =
P( Y(t_k) < y | Y(t_{k-1}), Y(t_{k-2}) )

That is, the probability distribution of Y(t) depends only on
the two points immediately in the past.

If Y(t) is stationary, Gaussian, second-order Markovian and continuous
in probability, then Y(t) would seem to satisfy a certain second-order
linear SDE [4]. Does anyone know a reference for a rigorous proof of
this fact? Thank you for your help!

Steve Finch
http://algo.inria.fr/bsolve/

References

1. J. L. Doob, The Brownian movement and stochastic equations,
Annals of Math. 43 (1942) 351-369.

2. L. Breiman, Probability, Addison-Wesley, 1968, pp. 347-351.

3. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic
Calculus, Springer-Verlag, 1988, p. 358.

4. A. Gelb et al, Applied Optimal Estimation, MIT Press, 1974.
.