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358 Chapter 10 Further Topics
Chung and Williams, 1990, Karatzas and Shreve, 1991, and Oksendal, 1992). The
solution of (10.4.2) can be written as
or equivalently,
X(t) e −at X(0) + σ
t
0
e −a(t−u) dB(u) + b
X(t) e −at X(0) + e −at I(t)+ be −at
t
0
t
0
e −a(t−u) du,
e au du, (10.4.3)
where I(t) σ t
0 eaudB(u) is an Itô integral (see Chung and Williams, 1990) satisfying
E(I(t)) 0 and Cov(I (t + h), I (t)) σ 2 t
0 e2au du for all t ≥ 0 and h ≥ 0.
If a > 0 and X(0) has mean b/a and variance σ 2 /(2a), it is easy to check
(Problem 10.9) that {X(t)} as defined by (10.4.3) is stationary with
E(X(t)) b
a
and Cov(X(t + h), X(t))
σ 2
2a e−ah , t,h ≥ 0. (10.4.4)
Conversely, if {X(t)} is stationary, then by equating the variances of both sides of
(10.4.3), we find that 1 − e−2at Var(X(0)) σ 2 t
0 e−2au du for all t ≥ 0, and hence
that a>0 and Var(X(0)) σ 2 /(2a). Equating the means of both sides of (10.4.3)
then gives E(X(0)) b/a. Necessary and sufficient conditions for {X(t)} to be
stationary are therefore a>0, E(X(0)) b/a, and Var(X(0)) σ 2 /(2a). Ifa>0
and X(0) is N(b/a, σ 2 /(2a)), then the CAR(1) process will also be Gaussian and
strictly stationary.
If a>0 and 0 ≤ s ≤ t, it follows from (10.4.3) that X(t) can be expressed as
X(t) e −a(t−s) X(s) + b −a(t−s)
1 − e
a
+ e −at (I (t) − I (s)). (10.4.5)
This shows that the process is Markovian, i.e., that the distribution of X(t) given
X(u), u ≤ s, is the same as the distribution of X(t) given X(s). It also shows that the
conditional mean and variance of X(t) given X(s) are
and
E(X(t)|X(s)) e −a(t−s) X(s) + b/a 1 − e −a(t−s)
Var(X(t)|X(s))
2 σ −2a(t−s)
1 − e
2a
.
We can now use the Markov property and the moments of the stationary distribution
to write down the Gaussian likelihood of observations x(t1),...,x(tn) at times
t1,...,tn of a CAR(1) process satisfying (10.4.1). This is just the joint density of
(X(t1),...,X(tn)) ′ at (x(t1),...,x(tn)) ′ , which can be expressed as the product of the
stationary density at x(t1) and the transition densities of X(ti) given X(ti−1) x(ti−1),